Angle Measuring Practice & Fine Motor Skills

My 10th grade Geometry classes missed two critical years of in-person schooling in middle school.

One thing I've noticed is that these students seem to have more trouble than I had anticipated, and one of the things they seemed to struggle with most is working with a physical protractor in 3-dimensional space. The idea of using a physical tool to measure a spatial object seemed very foreign to almost everybody.

Every time I encounter something like this in our post-pandemic world, I've learned to ask myself what impact distance learning may have had on the students who were stuck at home. My training, my experience, and my own research have taught me that our physical organism moves towards health, so long as we assist it. That makes me want to treat this problem not as a deficit of mind but rather as a gap in experience.

I realized I needed to create an activity that would backfill this gap in experience and empower students to move forward from where they are.

So here is my Angle Measuring Practice activity from today. There may be typos or my own silly measuring errors because I'm tired. 

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Start by printing and hanging angles #1 - 12 around the room. Kids at each table number (#1 - 9) start their measuring journey at their corresponding angle number. Everybody measures every angle. Table members compare measurements and call me over for a read through. We check for understanding -- did you accidentally start your measuring from 180 rather than 0? Clarify that. Support kids at measuring stations by asking/showing where the vertex goes. How do you align one side of the angle against the protractor?

Kids will start clarifying for each other. This is good.

When they complete the circuit, whole tables called me over for a check. We talked about estimation and levels of precision. 

Then I gave them level #2 with instructions. Now they have to check their own work, using what they know about linear pairs and the sum of their measures. 

When they finished, they did level 3.

I don't know what it is about hanging stuff around the room and getting kids standing up, but it works. By the time they finished the circuit of the room, they were deep into the work.

Physical collaboration is powerful. 

This reminded me to use it.

PART III: The Four Mistaken Goals of the Discouraged Child

 2. A Struggle For Power

Misunderstanding unpacked: "I only have value if I prove my own value by refusing to cooperate with the teacher and/or the classroom and its norms." 

 

Teachers who frame classroom management in terms of compliance/non-compliance are inviting a failure of classroom community. For one thing, there are 36 of them and only one of me. But more importantly, it is a missed opportunity to create the context in which a discouraged child can come to view herself as a community member.

 

As Dreikurs says, 

It is a grave mistake to try to overpower a power-drunk child. It is also futile. In the ensuing battle, which becomes chronic, the child merely develops greater skill in using [their] power and finds greater reason to feel worthless unless [they] can demonstrate it.  (Dreikurs, page 61)

 

Most teachers have heard that they shouldn't engage in a power struggle, but side-stepping the challenge to a power struggle activates reactions that run deep. In addition, if the child's intention is to assert power, any "attempts to make [them] stop only intensify [their] disturbing behavior." (Dreikurs, page 62). Resistance is easier said that done.

 

What I have found most effective is to build the habit of noticing within myself whenever a student is trying to engage me in a power contest. The secret is to notice the internal trigger, name it, and refrain from reacting with the conditioned habit they are trying to generate. 

 

It takes practice. I find myself mentally chanting, Notice... and refrain... notice... and refrain. 

 

And it works. This is the fastest way I know to defuse the power struggle most of the time.

 

A couple of notes. The first is, this doesn't mean there are no natural or logical consequences. There definitely are. And one of the things our students are learning is how to function in society. 

 

For example, one always has the power to not do one's tax returns, but then there are consequences. Some of those consequences are serious. But teachers should always avoid making the mistake of believing that we have to be the personal, immediate, and traceable agent of every consequence a student will ever encounter. That's just not sustainable, and teachers need to be in this work for the long haul.

 

If a student doesn't submit an assignment that is due, I simple place a zero in the gradebook with a comment that late work is always accepted, but is subject to a one-point late fee. Zeroes are erasable. Basically, it's the same as doing your taxes. Shit happens, and sometimes you have to turn things in late. If you learn to plan ahead, there's no late fee. The point is for the student to learn how to meet their deadlines and manage their life's competing obligations. Don't make it into a big deal. Think of yourself as the government agency in charge of tracking and reflecting students' work.

 

This year I encountered a weird new manifestation of this power struggle. Along with all the teachers at my school, I use Google Classroom as my CMS (Classroom Management System). My policy on homework is to simply check it in as the basic routine step. I ask students to turn in their homework by uploading a photo of the first page of their handwritten work. This is the honor system. Most students most of the time turn in a photo of their own detailed handwritten work as evidence of their effort. Since homework for me is only a record of deliberate practice with metacognitive self-reflection, this is enough. I can tell from students' Burning Questions the next day how deeply they have engaged with the work, and this gives me the formative assessment info I need to adapt my instruction.

 

Only later -- if there's a problem -- or occasionally -- as a spot check -- do I go through students' homework submissions in more detail.

 

But this year, I got a surprise at the end of the spring semester.

 

I noticed that one student had turned in a photo of a cheeseburger with fries in place of his homework for that day. In accordance with my policy, I put a zero into the points field and typed into the comments field, "This is a picture of a cheeseburger with fries. Please upload a photo of this assignment to receive credit, minus a one-point late fee."

 

No reaction. No emotion. I just switched into functioning as the conduit of natural and logical consequences for a decision that was made.

 

Of course, this experience encouraged me to look at his other submissions for the semester. And sure enough, I had missed some other cheeseburger submissions as well. Because I am acting as a neutral agency in this regard, I changed all of the scores for those assignments to zeros and copied and pasted my same neutral instructions into the comments field. There were also photos of a sneaker, a bicycle, and photos of somebody else's homework with their name printed at the top. I modified all scores for these and copied my neutral instructions into the comments field. Zero, zero, zero. 

 

Naturally, the student's grade started dropping precipitously, which finally prompted them to come up to me and apologize for the huge clerical mistake they had made. They asked if they could resubmit these homeworks. "Of course!" I told them. "That's the whole idea!"

 

In this case, the natural consequence of having to redo all those homework assignments was the need to spend time redoing them all -- knowing that I would look them over far more carefully than I might have done before, and also that I might discover even more phony assignments. 

 

A valuable lesson was received and integrated with far less conflict and more face-saving than if I had become emotionally activated. The lesson for me was, Don't bite the hook.

 

There were other students who had done the same thing as well, and I treated everybody who had done so with the most consistent standard of fairness I know. One student, who actually submitted photos of a table mate's work was horrified to learn that I'd sent that table mate and their parents an email notifying them that somebody else had been submitting photos of their work and that in California, this is considered academic cheating under the Education Code and would be a serious offense. Then the student learned that I'd sent an email to them and to their parents as well about the situation. 

 

Most students are good people. But they are also adolescents and they make ridiculous choices and mistakes. The horror for those two students of being caught out and unmasked for their parents was far more powerful than any rage-based consequence I could have meted out in the heat of emotion. And these two students both digested powerful lessons about the consequences of not living up to their own responsibilities. They each apologized to me personally and it was clear that they're not going to do this again. They were also grateful for the grace they were shown. 

 

Never argue with a power-drunk teenager. Find ways to notice and name the power move that don't jeopardize the underlying relationship between you unless you have absolutely no other option.

 

Our job is to support students in learning to rise to the occasion.

PART II: The Four Mistaken Goals of the Discouraged Child

 The Four Mistaken Goals of the Discouraged Child

 What I love most about Dreikurs' psychology is his clarity about boundaries -- both the parent's and the child's. By putting courage and encouragement at the center of his framework, he centers the child's development and the adult's cultivation of strengths and belongingness, not of weaknesses or deficits and punishment. As he says, 

Parental love is best demonstrated through constant encouragement toward independence. We need to start this at birth and to maintain it all through childhood. It is made manifest by our faith and confidence in the child as he is at each moment. It is an attitude which guides us through all the daily problems and situations of childhood. Our children need courage. Let us help them to develop and keep it. (Dreikurs, p. 55)

This framework applies just as much to the teacher's role in the classroom as it does to the parent's role at home. Applying his framework to my classroom management has been a lifesaver and an opportunity to generate meaningful connections.

 The centrality of courage in Dreikurs' model reframes misbehaviors in a constructive and workable way. For Dreikurs, it is important for adults to understand that the child who is misbehaving or not cooperating in some fundamentally important way is the opposite of encouraged -- this child is discouraged. In Dreikurs' framing, misbehavior is the manifestation of discouragement. The genius of this insight cannot be overstated. Discouragement is a workable condition -- one from which a child can heal and reconnect with the social fabric of belonging. 

For this reason, Dreikurs invested an enormous amount of energy in his research into understanding what it means for a discouraged child to be discouraged. Through this research, he identified what he called the "four mistaken goals" of the discouraged child. Understanding these mistaken goals makes it possible for an adult to learn appropriate, insightful, and very creative methods of responding that will redirect the discouraged child into a more productive approach to finding belongingness.

The four mistaken goals of the discouraged child can be summed up as follows:

1.     undue or excessive attention-seeking

2.     a struggle for power

3.     escalation of the power struggle into the pursuit of revenge and retaliation

4.     shutting down and giving up as a form of self-protection against further discouragement

Even though these are dysfunctional strategies for dealing with discouragement, they deserve acknowledgment for how brilliant and resourceful they are. But they are dysfunctional and we can help students to do better, both for themselves and for the whole classroom community. And this is where the framework of the four mistaken goals of the discouraged child offers highly effective ways of helping students find their way back into healthy belonging and connection.

I want to emphasize that understanding these categories isn't a panacea. Nothing will be instantaneous. But Dreikurs' methods provide a sane, bounded, and healthy lens through which to understand what is going on with these students and to reflect on meaningful ways to address it.

Here's how I experience these in my math classroom.

1. Undue Attention-Seeking

Misunderstanding unpacked: "I only have value if I receive individual one-on-one -- and often immediate -- attention from the teacher." This can take a few different forms. 

During collaborative mathematical group work (using Complex Instruction or other approaches), the student who insists on turning away from their table group and receiving help directly from the instructor is seeking undue attention. This is the reason why we keep returning to the Complex Instruction rule of "only whole-group questions" to the instructor. Part of what students are learning during mathematical group work is self-reliance and peer-reciprocal-reliance. Students are also building their capacities for executive function, self-regulation, and impulse control. Our goal for students is to help them become self-confident, independent, and self-directed learners. They are learning how to look inward to construct their own answers using the best tools and ideas they know, and to engage in positive, pro-social, and interdependent analyses and investigations when they run out of their own personal knowledge. We want them to learn how to exhaust their team's collective resources first before reaching out to the teacher because this is how healthy adults function in the outer world. This is a very different approach to authority than younger children take, because students' self-actualization learning goals are as important as their mathematical learning goals.

The purpose of the "only whole-group questions to the teacher" rule is to build a healthy student fluency in independent thinking, problem-solving, and self-regulation

Another way I encounter undue attention-seeking is during unstructured group work or classwork. When students run out of runway, I encourage them to come up and ask their questions. Over the years, I have become a master at asking questions that elicit the student's thinking and at crafting the tiniest possible hint I can provide. My goal is to help them access the framework they have and to give them a little boost that can help them break through their stuckness and get them moving to the next level. 

There are healthier and less healthy ways that students try to use this kind of access. The unhealthy ways of approaching this that spill over into undue attention-seeking occur when a student plops themselves down and tries and monopolize as much time and attention as they can access from me. This is where gentle redirection is so important. I tell them I will only dispense one hint or piece of help at a time, but I invite them to come back with their next stuck point as many times as they need to. I reinforce that they're welcome to come back with their next question if they need it. 

I try not to allow any one student to monopolize access to me as a classroom resource. This is part of building trust and also of building classroom community and positive interdependence. In my 7th block class this past spring, a new piece of classroom culture emerged, in which students talked openly about "sharing the wealth." The student who had come over to ask for help on, say, problem 11, became a kind of shared community resource. Another table would call this person over to ask for guidance and they would confer and share insights -- never simply doing the problem for each other. 

This was far and away the greatest high point of my teaching year.

Belongingness Comes First: Classroom Management through a Harm-Reducing Lens -- PART 1

 

This is a post that has been rattling around in my mind for a long time, but this has been the year when colleagues younger and older have asked me to please write this down. It's the first of a series of posts I'm going to do this summer, mostly to help myself remember what I need to know when I have forgotten what works. May it be of benefit to others as well.

NOTE: The book of Rudolf Dreikurs that has most deeply influenced my work is from 1964 and is called Children: The Challenge. While every cultural reference in this book may feel cringe-worthy and embarrassing to you, don't let that put you off. Dreikurs was a true master, and his framework and insights on every page ring as clear and true as the most finely tuned bell. It just happens to come from a different age. Don't be tossed away. Dreikurs' psychological methods and insights have a clarity you will not find elsewhere. Take what is beneficial to you and release what does not serve your needs.

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Belongingness Comes First

In the child's mind, belonging is a life-or-death question.

 

The fundamental insight of Adlerian child psychology -- and of Adler's disciple Rudolf Dreikurs, who originated so many of the parenting concepts we now take to be obvious -- is that every child is driven to seek out belonging. The behaviors a kid manifests are designed to achieve their survival goal of secure belonging. 

 

How can this be used in the classroom? Well, if a child enters a new situation and immediately experiences a sense of belonging, then things will tend to run smoothly. The child will read the room, unconsciously relax, and dive into the stream of seamless and happy participation.

 

Sounds easy, right? :)

 

In practice, it can be challenging to foreground belongingness in the classroom -- and to keep it in the foreground. It took me years to let go of my first teacher impulse to talk first and engage second. The way I've found best to implement belongingness is by consistently using a non-spoken daily structure that is impossible to ignore. When students walk into my classroom, the first thing they see projected on the screen is a "Welcome to Geometry!" slide with the instructions for the day. The very moment that class begins, I press "play" on my slide and the thundering drum fill from the Hawaii Five-O opening theme music crashes over the room. I have an ancient Bose speaker that amplifies the music. 

 

It's impossible to ignore. But just in case students manage to ignore it, when it fades out, I start yelling. "Instructions are on the board! Read them and follow them! Let's go! Let's GO!"

 

It's important that the first time they hear my voice, it is in service to our shared collaborative mission. This establishes the ground rules of belongingness in my classroom. WE have a job to do together. I'm just here to encourage that along.

 

As we move into the heart of the first week, I use this structure to train students on how to work with Burning Questions. A Burning Question is a question about the previous day's work that students can't answer for themselves. THAT is the proper use of the teacher. So the first segment of every class' instructions is to prepare for the Burning Questions segment of class. I'll take every BQ students have, but I don't accept the answer "all of them." That's lazy and threatens belongingness.

 

Once we've assembled our list of Burning Questions, we walk through worked examples, but the way I do worked examples is very different from what I've observed in other teachers' classrooms.

 

As Rudolf Dreikurs says, "We must observe the result of our... program and repeatedly ask ourselves, 'What is this method doing to my child's self-concept?" (Dreikurs, p. 39). As teachers we are always faced with the choice of encouraging independence, self-respect, and sense of accomplishment or undermining it. A huge part of what children are learning to develop through productive struggle -- psychologically speaking -- is a healthy ability to tolerate and manage frustrations. Obstacles are a critical fact of adult life. We need to support children in developing the courage to see productive struggle as another texture of adult life they can learn how to face and overcome. As Dreikurs puts it, encouragement -- not praise -- plays a crucial role in helping students develop the "self-respect and sense of accomplishment" they will need to find their place in our world. (Dreikurs, p. 39)

 

Burning Questions is a Narrated Thought Process

The key to encouraging students' courage during Burning Questions is not to do any part of the problem that students can do for themselves. Burning Questions is a profoundly interactive segment of class.

 

In practical terms, this means that demonstrating worked solutions requires modeling the metacognitive questions I would ask myself as a learner when I encounter a math problem of this type and find myself stumped. Modeling courage is essential. Students always already know something. And since what they know is the best thing they know, I use my understanding of their ZPD (Zone of Proximal Development) to find a simpler starting-point question that they can answer. This is usually a question about identifying the situation at hand in the problem. What kind of triangle do we have here? Do we have parallel lines? What kind of angle pair are angles 1 and 8 in the diagram? Do we have an altitude-to-the-hypotenuse situation here?

 

Encouraging students to name -- and use names for -- different mathematical situations is a critical part of my pedagogy. It enables me to ask them dozens or even hundreds of times whether we can spot one of our familiar important mathematical situations

 

I  break down and ask the questions; if the students don't provide the answers to my much-simpler questions, then we simply don't progress. My wait time game is strong. I can sit silently, blinking, for three whole minutes, if need be. 

 

Belongingness dictates students' need to collaborate to find answers. I name and narrate behaviors that I see which are positive and constructive learning behaviors which everyone in the room can do. "I see some people flipping back through their notes, looking up different situations. That seems like a good idea to me." More pages start getting flipped. Quiet conversation ensues at different tables. Students point out possibly relevant parts of their previous days' notes.

 

Eventually somebody brave will pipe up with an idea. I will repeat the idea for the whole class and ask if that makes sense to them. I will often take a vote. I am not some deified source of right and wrong answers. I am actively trying to encourage them to rely on courage and on each other. We'll take a vote. Only then will I confirm whether or not this makes sense.

 

Wrong answers are fine and we honor them by interrogating them and passing by quickly. They give me an opening to ask more clarifying and refining questions about key properties and distinctions. I tell students that spotting known mathematical situations is like bird-watching. You need a field guide and practice identifying the distinguishing characteristics of different situations. This is how people learn.

 

This is my process for modeling courage and resourcefulness during productive struggle. Mine might work for you, or you might need to find your own on-ramp.

 

Students get a lot faster at this interactive process. At the beginning of the school year, I may have to wait minutes before moving on. Within a few weeks, it will only take a matter of seconds for students to pipe up with answers for each questions.

 

"What kind of situation do we have here?"

"Altitude to the hypotenuse situation."

"Good. What pieces of the situation do we know? Which lengths do we have? Which lengths do we need to find?"

 

Never answer a question that the students can answer for themselves or for each other. This is how we cultivate courage and endurance for productive struggle.

 

Piece by piece, question by question, we walk through the problem together. Belongingness is non-negotiable. Whole-class segments not only teach participation and collaboration skills; they enact belongingness. Even if you are totally off-task, absorbed in texting, or feeling heartbroken over your relationship break-up or something even worse, during whole-class segments, you still belong.

 

As Dreikurs puts it, "All comparisons are harmful." (Dreikurs, p. 44)  Whole-class segments are not the time to yell at a kid for being glued to their phone. If you have to have that conversation with a kid, do it in private. "Her abilities will increase only if her confidence is restored." (Dreikurs, p. 44). The ultimate larger goal is to encourage each student to reach "the point where [they] will enjoy learning... and may find out how much more capable [they] are than [they] have thought till now." (Dreikurs, p. 45)

 

The Enemy of Belongingness is Discouragement

 

But if a child has become discouraged, their focus will shift from participational and cooperative behaviors toward less constructive behaviors that are unconsciously designed as defense mechanisms to protect them against their perceived failure to achieve belonging.

 

This is the most important insight a teacher can integrate into their classroom management orientation. 

 

These are not deliberate strategies. These are observed and catalogued patterns of behavior that arise when a child fails to achieve successful belonging. 

 

This is good news for classroom teachers. If you can make sense of and spot these behaviors, you can generally find solutions for redirecting them into pro-social behaviors that will be more satisfying for everybody involved, including both you and the child.

 

When I first read Dreikurs' analysis of the four mistaken goals of the discouraged child, I had a breakthrough in my understanding of students' misdirected behaviors. I'm writing this down because I want other teachers to be able to make sense of this too.  It's so much easier to deal with when you have a validated framework for making sense of what's going on.

 

Here is Dreikurs' 30,000-foot perspective, with one modern update from me in brackets:

 

Children want desperately to belong. If all goes well and the child maintains his courage, he presents few problems. He does what the situation requires and gets a sense of belonging through his [success] and participation. But if he becomes discouraged, his sense of belonging is restricted. His interest turns from participation in the group to a desperate attempt at self-realization through others. All his attention is turned toward this end, be it through pleasant or disturbing behavior, for, one way or another, he has to find a place. There are four recognized "mistaken goals" that such a child can pursue. It is essential to understand these mistaken goals if we hope to redirect the child into a constructive approach to social integration. (Dreikurs, p. 58)

 

Once I started understanding student misbehaviors as falling into one of the four mistaken goals of a discouraged child, it became much easier to find appropriate and effective methods for redirecting student energies in healthier and more constructive ways. 

 

I want to emphasize that there are two important elements to healthy classroom management here. One belongs to the teacher, keeping a clear understanding of the situation, not allowing oneself to get triggered, refraining from reacting to the triggering behavior, and maintaining healthy boundaries to preserve your own sanity. The other belongs to the student who is suffering and acting out in one of the four mistaken ways. The student needs to feel seen, understood, appreciated, and guided in a healthier way. 

 

But because the student is still developmentally a child -- with a child's incompletely formed sense of judgment and executive function -- plus whatever other factors are at play in the child's outside life situation, this is one of the most important situations in which to understand that telling is not teaching. This is the real genius of Dreikurs' approach. He understands that the language of actions is the only way in which the adult can successfully reach the child with these messages. It has to take place at an unconscious level. And it is by taking this approach and following it all the way through that a teacher can reach and encourage the student to follow the better, more effective and healthier path.

 

To put it another way, we have to use psychodynamic wisdom in order to achieve psychological goals.

Choices Have Consequences: How the 9th Grade Failures at Lowell Shine a Spotlight on SFUSD's Literacy Emergency

By Megan Potente and Elizabeth Statmore

The news coverage of the rise in 9th grade Ds & Fs at Lowell High School after this first year of lottery admissions fails to mention two things: the fact that in a wealthy city that prizes equity, San Francisco Unified School District has been promoting an unacceptably high percentage of 8th graders who cannot read at grade level; and the fact that the sudden change in Lowell admissions is what is shining a bright light on these disastrous reading results.

A recent audit of SFUSD’s K-5 reading instruction program shows how the district’s toxic love affair with debunked reading fads has been harming students in predictable ways. This Lowell 9th grade class is the first cohort not prescreened for academic competencies; therefore, they must be seen as representative of future incoming cohorts of Lowell students under a lottery system. So these troubling 9th grade results this year at Lowell promise to become the new normal for future Lowell cohorts of students who will be randomly assigned to Lowell. 

The tripling of Ds and Fs in one year was bracing to us Lowell teachers. Reading is a skill built on foundations, but the literacy audit revealed almost nonexistent teaching of reading foundations. 92% of SFUSD classrooms were found to be not meeting standards in this area. Reading fluency depends on efficient and accurate word reading, which needs to be taught in K-2. When teaching doesn’t prioritize these fundamentals, kids move into the upper grades without grade-level fluency. As a result, reading is difficult, which means struggling readers tend to read less, and consequently their vocabulary and language development suffer. The cumulative impacts of poor early reading instruction are astounding.

As a district, we are now reaping the results of our poor curriculum choices. The most recent pre-pandemic data show that 55% of all SFUSD students don’t meet standards in English Language Arts (ELA), and there are huge gaps in the performances of specific subgroups. Only 21% of Black students met ELA standards. The results were even worse for English learners and students with disabilities.  

When Lowell teachers first noticed the high number of Ds and Fs our students were earning, we did what good teachers always do: we compared notes. Who was thriving? Who was struggling? We analyzed student work. We scoured cumulative files and lexile reports. We used student data to inform instruction.

But one thing stood out: none of us had ever seen so many 9th graders at Lowell struggling to read at grade level. 

We were shocked by this lack of reading readiness. Could this be a snapshot of the general level of 9th grade reading readiness across all of SFUSD?

Judging by the audit report on SFUSD’s early reading program, it certainly seems possible.

True Equity Demands Improving K-5 Literacy Results
SFUSD leadership needs to accept that the Ds and Fs among Lowell 9th graders this past year are a wake-up call – evidence that calls for a return to evidence-based reading curricula in K-5. SFUSD can no longer afford to ignore the science of reading -- a field whose consensus is so broad it has come to be called literally ‘THE Science of Reading,’ 

The continued use of debunked literacy methods in the K-5 years has taken its toll on all our students. These 9th graders at Lowell are just the canaries in the coal mine.

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Elizabeth Statmore is a math teacher at Lowell and an executive board member of Families for San Francisco. Megan Potente, a 20-year elementary educator, now serves as co-state director of Decoding Dyslexia CA. She is the parent of an SFUSD graduate.

Why I'm In This Fight - the Future of Mathematical America

 Jo Boaler is utterly missing the point.

My argument is simply, Enough is enough.

We need all of our children to be mathematically competent so we can get their voices and minds into positions of leadership. We need our Black and brown children to achieve the levels of mathematical competence they will need to get them into leadership and to fulfill the promise of their brilliance.

We need all our youth to be mathematically well-educated and well-prepared.

We need them to have an effective mathematical education.

And we need to be using facts and metrics that reflect reality, not magical thinking.

This requires that California stop merely rearranging the furniture of the high school math framework every eight years and start pouring some of our energies in to addressing the actual problems that are preventing our historically least-reached students from entering middle school with a healthy and solid foundation in early childhood and elementary mathematical capacity.

We need appropriate developmental mathematics from early childhood onward. We need it NOW. Mathematical play leads to mathematical curiosity. Curiosity leads to intellectual hunger and to rigor, precision, and competence.

I want us to wake up and work together flood the world with diverse, well-educated high school graduates armed with mathematical competence because THAT is the foundation of accomplishing great things in the outer world. 

THAT is what's going to save civil society and our planet.

I read accounts like David Brooks' latest report about the future of right-wing America and it puts me into a cold sweat.

These people are training their children to take over.

It's time we on the left and in the center start taking that threat seriously.

That threat is an existential threat to our children, which is what gets me up in the morning to teach my high-achieving, high-poverty students to become mathematical warriors. I want them to be well-equipped to do battle with these people. 

I wake up every morning and work to help build the world that OUR children have the vision and brilliance to create -- not the world that those children are being equipped to create.

It's time for us to push past the first wave of naive, solipsistic navel-gazing of "reform mathematics."

Our underserved youth are tougher, smarter, and hungrier to master mathematics than Jo Boaler realizes.

It's time to move on from what doesn't work and hasn't been properly measured.

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.