cheesemonkey wonders

cheesemonkey wonders

Sunday, May 30, 2021

Piercing the Enchantment

Having survived more than a full year of online learning, I am readier than ever to return to in-person learning. 

I've been thinking a lot about what I've learned over the past year and what I will carry forward with me into the classroom as we return to in-person learning.

As a person with a lifetime of getting hopelessly lost, I have learned that once you notice that you are lost, the first thing to do is to just stop. Stop and get reoriented. This is one of the things I have learned through the power of meditation. When we meditate, we sit down and stop paying conscious attention to the crazy stories our panicked monkey minds are trying to tell us. We anchor our mind in our breath and just stay there. When the river of thoughts delivers another raging dumpster fire of crazy thoughts, we notice it, label it, and disidentify from it. Huh. That’s interesting. Another dumpster fire worth of crazy thoughts. And each time, we quietly return to anchoring ourselves in our breath. 

The more we do this, the less power these storylines have over us. As Suzuki Roshi says, we start to understand that what we refer to as "I" is really only a swinging door. Breath flows in, and breath flows out. When we anchor our panicking minds in our breath, we return to the safety and goodness of the present moment.

When I teach, I anchor each lesson in an Essential Question. Under ordinary circumstances, my Essential Question is always some variation on the meta-question, Why do I believe this is worth your time and attention today? But these have not been ordinary times. Instead of the usual four-plus hours a week of math class I have always had, under distance learning, we have had no more than an hour and a half of teaching and learning time together each week – for the whole week. Instead of synchronous time, we've had to make do with asynchronous learning experiences, which can be isolating, discouraging, and frustrating.

This has forced me to rethink my entire concept of the Essential Question for my classes. It's hard to keep the momentum going when you lose that day-to-day in-person connection. Many students reported feeling so alone without the daily contact of in-person schooling.

And so to help them – and to stay grounded in my efforts to help them – I changed the focus of my classes. 

I began to think of my class as an Essential Anchoring Place for students first and foremost. An anchoring place where we anchor our minds using math.

My Essential Question for each day turned into something like this:

How can I guide your attention to some things that can help you when you feel absolutely and utterly lost?

When you feel lost, the first thing to do is to stop. Just stop. Stop moving, stop striving, stop efforting. 

Just sit the heck down.

Reconnect with the body, with the breath. Return to the ground.

In meditation, we sit down and anchor our minds in our breath.

In Classical mathematics, I realized, we anchor ourselves in definitions. Definitions are mathematical bedrock. We bolt ourselves to them, making a conscious decision to take them to be true and move on from there.

We use definitions to orient ourselves. We do this not because they were handed down on stone tablets from Mount Zion or Mount Olympus but because for thousands of years, human thinkers have decided – as a thought experiment – to take these to be true and explore what can happen next.

These are our assumptions, and we acknowledge them as such.

And just as a map enables us to construct a working mental model of our journey, a mathematical definition gives us a working mental model of mathematical reality. A map is a tool – a good-enough humanly constructed tool that encodes our best, most reasonable, working understanding of how the world fits together. The journey we are on is a relay, and the maps have been handed down across generations for thousands of years. We are only responsible for our portion of the journey, though we inherit both the tools and the biases they encode.

Maps are cultural artifacts – texts which are products of the terrible racist systems in which they were constructed. They may contain some of the best thinking people were capable of, but they are also encoded with some of the worst, most wrong-headed, and most biased thinking of the dominant cultures in which they were developed. 

This is why we take them only as heuristics. They are imperfect pointers to a truth, not the truth themselves. As the Buddha often said, “My teaching is like a finger pointing to the moon. Do not mistake the finger for the moon.” 

So just as a map guides our thinking about how we journey in the outer world, a mathematical definition helps us to take mathematical journeys in the inner world of our thoughts and minds. 

Students learned that our foundational mathematics are built on definitions. We do not prove these – we take them to be true. 

Definitions provide an on-ramp for a crucial way of thinking in mathematics: the foundation of thinking in conditions. What is necessary and sufficient for a figure to be considered a circle? We start with its Classical definition: a figure is a circle if and only if it is the set of all points in the plane equidistant from a given point, the center. Where does it start? With a center. Does every circle have a center? Yes it does – by definition. What else does this definition tell us a circle has? Students fasten on the idea of a fixed and equal distance. Is the circle a set of points? How many points? Does the circle as a mathematical figure include the points inside the figure? How do you know?

The definition becomes my students’ friend. It contains a set of tests. What if one point of the figure were discovered to NOT lie in the plane? Would it still be a circle? Why or why not? How would you know?

The idea of a set of true-false tests becomes a foundation in which students can ground their thinking. They always know something, and if worse comes to worst, they can go all the way back to the ground of the definition. In this way, mathematics becomes a tool for getting yourself oriented. It becomes a culture and a community of belonging. We have to look beyond ego, beyond personalities, beyond individual likes and dislikes, to uncover what is true, enduring, unshakable. Definitions open a door to ways of thinking that have proven themselves to be durable and useful over time – over years, centuries, millennia. 

We really go all Platonic in our search for definitions, seeking out the perfect and ideal mental forms. We developed a crazy love affair with thinking in conditions. What are the necessary and sufficient conditions for an object to qualify as a member of this category? We start with real numbers and the real number line. We unpack the definitions of positive, negative, and zero. We go a little Aristotelian for a moment. Every real number has a fixed address on the real number line, and all addresses on the real number line fall into one of three categories: positive, negative, or zero, which is defined as being neither positive nor negative – the perfect inflection point. What does it MEAN for a number to be positive? to be negative? to be zero? How do you know? Slowly we construct an answer – these are by definition. They are what we are choosing to take as being true. 

This leads us to another mathematical love affair – the habit of thinking in cases. What is the set of all possible cases here? Is this a possible case? Why or why not? How do you know?

Students do a lot of casting votes in the chat window. “OK, does this figure meet all of the conditions required to qualify as a circle – yes or no? Don’t hit return until I count down.” I give them a moment. “Three, two, one, zero – hit return.” 30 votes pop up in the chat window. Yes, yes, yes, no?, yes.

I run an anonymous survey to see how things are going. Students tell me, “I like this class.” “This class feels the most normal.” I’m surprised. My Zoom policy/default is cameras-off. Many students were too self-conscious about their living situations. 

I worked with that. Every day starts just like my class would start in person. Slide with instructions, homework, and a countdown timer running for two minutes. Hawaii Five-O theme music playing loudly. I mostly used my iPad and Apple Pencil to do what I would have done on the smart board or document camera. More direct instruction than I would like because we have only 30 minutes together three times a week. Camera transitions and breakout room logistics eat up too many precious minutes. I give up after the first two weeks.

How do we reorient ourselves when we feel hopelessly lost?

We stop, sit down, and think about what a question is actually asking us. We allow ourselves to wonder what tools we have that could help us find the good-enough appropriate next step.

And then we do it again.

Monday, May 11, 2020

With sincerity and love to Nikole Hannah-Jones and all the beloved urban public school parents struggling to make sense of learning under quarantine

Dear beloved urban public school parents,

Please forgive me for being out of touch. I know you're worried about your child's learning during the pandemic, and I know I've been remiss in explaining everything I've been doing since March 7th. You could say that I've been building the plane while I'm still learning how to fly it. Plus I've been so busy trying to figure out how to salvage what I can of my school's and students' year I haven't had time to communicate it all clearly to you and to answer your many questions -- which, to be honest, are also a lot of my own questions.

Apart from supporting our students and their families who are getting sick and dying, or losing their incomes and housing, or struggling with food insecurity, one of the top three biggest problems we teachers are wrestling with right now is this:
All of our best teaching & learning practices of the last 50+ years are based on layers and layers of assumptions of direct, in-person collaboration.
All of the most effective pedagogy is based on conclusive evidence that effective learning is socially constructed. We've trained at least two generations of teachers based on this assumption, and our entire schooling system is based on this premise. So we teachers are working on figuring out what we can salvage, given what is possible given our limited time and circumstances.

Collaboration is not only baked into the physical circumstances of our schools, it's also baked into the state-approved approved curricula.

Now under this pandemic -- and for the first time in about 3,000+ years of teaching and learning -- we are all physically distanced. In addition to being physically separated, due to the inequities in socioeconomic, housing, and medical circumstances, much of my urban students' learning has had to go asynchronous.

In all of recorded human history, this has not happened before. Everything we front-line teachers and administrators know and have learned about best practices being collaborative and cooperative is just gone.

So for me, as a veteran classroom teacher, this leaves me with three Essential Questions that no existing education "expert" can actually answer for me.
1) How do children actually learn under these circumstances?
2) How do I triage and reprioritize elements of the curriculum to accommodate this new teaching & learning reality and to ensure the legally mandated Free and Appropriate Public Education (FAPE) to which each of your children is entitled --including students with disabilities and learning differences?
3) What adjustments do I need to make to equitably engineer the greatest possible learning opportunities for all students, given the already-vast inequities that are being amplified every day in every dimension of this pandemic at every turn?
I'm spending much of my time working on these questions and talking to my professional teaching & learning communities so that we can address the new reality for as long as it holds.

Sorry to have been so out of touch. I promise I'll keep trying to do better.


Doctor S and every other public school teacher everywhere

Sunday, January 19, 2020

Rooting Out Opportunity Hoarding and Perverse Incentives in the Math Classroom

tl;dr   The incentive structure of the math classroom is broken.

I live and teach in a community where opportunity hoarding is rampant. Students hoard points as if they were drops of water in the desert.

This leads to some perverse behaviors in the classroom. Students who have mastered a task or level want to take their attention to other parts of their lives. Their attitude is, I finished MY work; therefore MY obligations to math class are done. Students who have almost mastered a task or level of a topic become demanding of my attention in infantile ways. As soon as they run out of ideas, they tug on my sleeve, demanding that I re-teach them (or re-re-teach them) individually or in small groups. They value productive struggle only up to the point where they get stuck. The most challenged students feel so ashamed that they don't even know how to get started or even minimally unstuck that they try to hide in plain sight.

In a word, the incentive structure here is truly broken -- and perversely so.

I believe this is because the incentives here are all based on an assumption of individual attainment.

To allow a culture of individual attainment (what score /grade/mark did I get?) is to be complicit with the toxic culture of opportunity hoarding that pervades our whole society. I believe that the drive to hoard opportunity is one of the most powerful factors underlying the culture of systemic racism and oppression in schools.

Dylan Wiliam talks about how feedback needs to be more work for the recipient, yet every working classroom teacher I know knows that you can't force a kid to read or digest the comments. This is especially true when you have massive classes. With 37 kids per class, it's just not feasible. Kids look at the score and move on.

In my view, this is because the incentive structure of the math classroom is wrong. Not only is it wrong, it is sick and toxic. And we need to rethink these incentive structures if we truly want math class culture to heal.

If my grade means I personally have mastered or not mastered a topic, then once I get the score I want, my job is 100% done.

My problem with this is that, from the societal perspective, that is not my job as a classroom teacher.

My job as a classroom teacher is to get everybody over the finish line at the highest possible degree of mastery. For this reason,  my classroom's economy of achievement needs to become more collective, and less individual. I need to cultivate an incentive structure of positive interdependence -- "I" don't win unless others win too. Then we all win together.

There are times in my room when we're 37 individuals and there are other times when we are one classroom community. This is how things work on teams and in organizations throughout one's life in the U.S. So if we're one classroom community, then we need every individual to be as empowered as possible to achieve at the highest possible level.

For this reason, I've been expanding my whole-class skills quizzes. For a compound, complex skill such as solving a multi-step special right triangle problem (with interdependencies along the way), the quiz that I give is one that individuals take but each person's grade is an average of the scores of all the individuals in the class.

For two days leading up to the quiz, we do intensive collaborative work, including reciprocal doing-and-teaching practices such as speed dating. We also have unstructured time in which students identify as tutors or learners and then work to help each other improve the overall level of mastery in the room.

Our goal is a whole-class goal of mastery -- not an individual one. The goal is to raise the overall level of mastery in the room. Our goal as a class is to get everybody's level of understanding up. If you want to sit off to the side and work on your chemistry homework, then you're going to have to answer to your peers -- not to me. And if you don't like the grade that the whole class achieves, then too bad. Positive interdependence rules the day.

There are always one or two students who are so addicted to the toxic culture of individual attainment that they object, demanding, "If I understand it and they don't, then why should I be punished?"

And I have to explain to them over and over again. I tell them, "That's an infantile perspective. The better-prepared everyone around you is, the richer and more powerful your own learning experience is going to be -- both now and into the future. My job is to provide you with the richest possible learning experience so that you can go as far as you want to go. My job is to set the floor, not the ceiling. And this is how I, as the expert on learning, am empowering us to raise the floor of understanding."

Our school is unusual in that students get to choose their classes, their sections, and their teachers. My classes are very popular and are always among the earliest to fill up.

I choose to use this platform and my privilege to educate them. I'm blunt with the students who complain. "Listen," I tell them. "You chose this section. If you'd prefer a teacher who only gives individual scores on everything and lets you work on your chem homework when you're done, then we should talk to Counseling and get you into a course section where your desires are going to be met, because that's not going to happen in my class. There are plenty of other kids who'd be happy to switch with you."

I realize this may sound harsh, but they usually come around. And the fact is that my job is not to give them everything they think they want but to teach them and help them get aligned with the reality of things as they are.

The results bear this out. The lowest average on this first whole-class score of all my Geometry sections was an 87. The highest was 93.7.

The number of "free points" I provide in other parts of my class (through professionalism, home enjoyment packet completion, etc) makes this a wash. Nobody's grade goes down because of anybody else, but most people's grade do go up because their understanding improves. And as I tell them over and over and over again, what they need to do to raise their grades is to improve their understanding. The structure of the whole-class skills quiz empowers them to do so.

There's also less cheating and more cooperation because the incentive structures are aligned with our better, saner values.

There is still a place for individual attainment. Unit tests are individually graded as is the final exam. But individual attainment is demoted in my classroom and is put into better balance within our classroom community.

Individual attainment and opportunity hoarding are symptoms of our society's sickness. If we want to heal our learning environments and improve outcomes, we need to be open to revising the unconscious, unspoken incentive structures that keep reinforcing the systemic oppression we need to heal from.


@KarenCampe asks:
Wow this is amazing. Kudos to you for implementing something that really changes the game.
Do you have parent pushback?
— Karen Campe (@KarenCampe) January 19, 2020

I'm fortunate to have a lot of support from both site and district administration. In my view, this is a moral choice. My job is to create an equitable learning environment. If a parent were to insist on an inequitable learning environment for their child, I'm not sure what there is that we could do to satisfy them, given that this is public education.

Thanks for the question.

@timteachesmath asks:
Thank you for sharing! 
You've detailed your conversations with those 'done early'; 
what do those still learning think? Is there pressure to catch up, or a super supportive community?
— Tim (@timteachesmath) January 20, 2020

They appreciate that there is time and support being made for them to master what they find challenging. They want to learn the skills, but they get to do so in a way that does not punish them for needing more time or practice. They appreciate being part of the solution rather than part of the problem. And they are better able to participate and achieve their ends -- which is the goal. We are trying to normalize high achievement for everybody -- not sort out who "got it" first and who didn't.

Thanks for the question.

Saturday, January 4, 2020

How do we teach our students there are other ways to interact with the world beyond permanent war?

   My last encounter with physics did not leave me with a deep confidence in the practicality of math or science to save us. The course was taught by a man with no practical skills or insights or interpersonal skills, even though he was a tenured full professor at Princeton. What came through was that this was a man who allowed his wife to cut his hair using an upside-down bowl as a cutting guide. His hair was never even mildly symmetrical.

   I put my faith in medieval literary history instead.

   During the Dark Ages, clusters of monks in far-flung Irish monasteries kept the fires of learning lit. While the Vandal and Viking hoards stole, looted, burned, sacked, and traded away every last good thing the city-states and peoples of Western Europe had built, the Irish monks in remote scriptoria copied and illuminated manuscripts that preserved and spread the greatest learning of the day. And they taught their new generations how to carry out these vital matters of preservation and transmission along the way.

   When everyone else was taking cover and hoarding, the Irish monks kept learning alive, so that when the need – and demand – for it reawakened, it would be ready. Their system was like a beehive.  When it became possible again, the hives could be opened again and the contents could be used and shared for the public good.

   The desert is a lot like this. Things appear to have gone dead on the surface, but just below the veneer, the Earth is teeming with life – positively giddy with abundance.

   This is what gives me confidence to keep teaching and learning.

   It gives me confidence that something will survive until there is intelligent life in our world and in our government again.

    How do we keep the fires of learning lit in our society while those in power all around us seem to be losing their minds?

   We do it by putting our faith in our teaching.

   We do it by banding together – and by not letting go. We develop a hard, hard crust and we protect our water resources well. We do it by remembering that our job is to stay present with our students and teach them how math and science are opportunities to understand the divine. We remember that human beings at our best are thinking-based life forms. We remember to bond with our kids. We remember that the kids are always watching and that we have an opportunity to model the next right thing to do. We do it by remembering that teaching and learning are effervescent and holy.

Monday, December 16, 2019

Purpose and Meaning in a Final Exam

Ever since @CmonMattThink tweeted out this poster that I'm just nuts about, I've been thinking about all the different uses I have for this saying:

Today, as we're starting our final exam week, I'm thinking about it with regard to assessing the meaningfulness of a final exam. 

We give common departmental finals, which is why, when my normally independent 9th graders felt compelled to pepper me with questions about the majority of questions on the test, it made me notice and wonder about the test itself:

Is this test a scavenger hunt for right answers? Or does it measure students' ability to provide evidence of their understanding?

There was a previous conversation on Twitter a while back in which some of us were debating the allowability and rationale for allowing students to have a reference sheet on a test. Darryl Yong (@dyong, who you should follow if you're not already) said something that I 100% agree with: if I am measuring higher-order thinking and problem-solving, then THAT'S what I should be measuring and a reference sheet makes sense.

If I am measuring students' effectiveness at memorizing things (such as vocabulary terms), then a reference sheet doesn't make sense because memory and recall are what are being tested.

So right now I'm sitting in a ditch, frustrated by the fact that my Algebra 1 students have been run off the road because the current test is merely a scavenger hunt for right answers to memorized algorithms... when what I REALLY would like to be measuring and understanding is, Do my students know what to DO with their knowledge in both routine and non-routine situations?

And I hate this particular ditch.

Wednesday, October 2, 2019

Building a Feel for 'Major Moves' in Proof

This year I'm experimenting with developing students' intuition for and sense-making about what we call 'major moves' in proof.

Rather than ask students to buy into the illusion that two-column proofs emerge spontaneously and fully formed from their brow, we are inquiring into how we mortals can better brainstorm and use our reference tools to create the sub-assemblies that we can use to build our rough draft proofs. Then we'll be better able to polish our final proofs and present our work.

This has meant that we are developing students' intuition that that these sub-assemblies are knowable and predictable. We call these our "major proof moves." Some of our major categories of major proof moves include:

  • the relationships between parts & wholes
  • a sense of bisectors and "half-ness"
  • parallels and the results of parallels
  • perpendiculars and their results
  • right angles and their results.

It's working out surprisingly well.

Today we started experimenting with using these higher-order concepts to work on harder, multi-stage proofs. The kids were quite excited to be able to figure things out.

Every year I am amazed at how many times students have to repeat an experience before they get that "click." This is giving us a much wider field to wander in as we master the art of proof.

Friday, August 30, 2019

The First-Ever Block 5 Math Department Pot Luck Lunch

Today we had our first-ever block 5 Math Department pot luck lunch.

We have a very large department (24 teachers) and not all of us have the same block for lunch. The block 6 lunchers had a pot luck last week, and so fueled by the competitive spirit, the block 5 lunchers were mobilized by one of the least social people in the department to host our own pot luck. The sign-ups were on the corner one of the office white boards, so the menu was shaped over the last week. And since we have by far the largest crew, hopes were high that we could pull this off. And we did so -- with style.

I made the Lemongrass and Ginger Roast from the Field Roast cookbook because I've been wanting to try it and we have a surprisingly large number of vegans and people with significant food sensitivities. I rushed home from school yesterday so I could make it and set it on the stove to simmer. Sarah made a salad which I had been planning to use as a base for my new favorite school lunch (chef salad surprise). Ernie made an incredible, silky hummus and pita bread. Lisa made a shrimp and avocado ceviche. Raymond made a wonderfully spicy red lentil dal, which I still don't understand how he heated up but it was delicious. Tyler brought donuts. Scott baked a boule of crusty sourdough with cheese and sausage.

Not to scale
But the runaway winners of the day -- and in my opinion, the offerings that raised this pot luck to the level of Artistry -- were Robert's mother's fried chicken wings and Alex's made-to-order waffles.

Alex brought in his waffle iron and a killer cinnamon syrup that he had discovered on the internet. His TA worked the waffle-making station at the standing desk, making waffles to order and generally supervising (don't worry, we fed her).

That's right.

We. Had. Chicken. And. Waffles For. Lunch.

In. The. Math. Office.

And fresh-baked sourdough bread. And vegan charcuterie. AND DONUTS.
Who can turn down a good donut?

At one point, Art strolled in from the computer lab and felt bad that he hadn't brought anything. Everybody jumped in and chided him, "DON'T FEEL GUILTY -- WE ARE DROWNING IN FOOD." So he loaded up a plate and joined us.

I don't know why we never did this before.

As we were winding down, Lisa said, "I kind of have a math problem I wanted to ask about, but I don't want to spoil it."

And everybody jumped in again and said, "DON'T FEEL GUILTY." She sketched the problem and most of us started tinkering on scratch paper, but as usual, Robert saw straight through to the core of the simplification. We all put down our forks and pencils in shock at his surprising yet unsurprising clarity.

Then, of course, we needed another round of waffles.

It was the best community-building activity I've ever done. And it was basically free. I hope we do it again soon because Raymond's grandmother made these killer dim sum things for our end-of-school pot luck last year that nobody knows the name of but everybody devoured them. They look like little fried footballs with some kind of mystery meat or veggies inside their cavernous pockets. And they are TO DIE FOR.