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.

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FOLLOW-UPS:

@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.

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

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

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

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

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

Your face here

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

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

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

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

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

Here's an example: how I use Talking Points both before and *for* mathematical conversation

OK, here's an example of how I used Talking Points first to get students primed for listening and considering other viewpoints, and then to get them to listen to and consider other viewpoints that can cause them to change their minds.

As our first activity following our first test of the semester, we did these Talking Points to start class.

These talking points were not especially successful, but they opened the door for the similar triangles discussion that followed.

We debriefed a bit, then I handed out this lovely, subtle activity from Park Math (Book 3, #20), and I asked them to change (a) to become a Talking Point, as in, "Triangle PRQ is similar to triangle STU." They were, as always, charged with doing three rounds and justifying their opinions.

Ten minutes of conversation ensued.

Next, I wrote three headings on the whiteboard (Agree, Disagree, Unsure) and asked each table in turn to tell me which conclusion they had come to and why. One by one, I wrote the table numbers under the categories where they located themselves (Agree, Disagree, Unsure).

And I held my tongue as table after table disregarded the order of vertices to tell me that, Duh, of course, they are similar triangles. I held my tongue because I trusted the process and had a felt sense that in a room full of 37 people, surely SOMEBODY would express a different, correct opinion.

And lo, it came to pass.

Table 6 bravely offered their belief that the triangles named were not similar because the order of vertices in each was not corresponding.

And one by one, the little lightbulb moments popped around the room.

I kept the discussion going until we were through with all 9 tables. Then, and only then, did I give tables another round in which they could change their opinion about what was actually going on in the diagram. 

Afterwards, we discussed what had happened. What did happen, I asked them. And they responded that something they heard made them realize they wanted to change their minds.

So that was my perfectly imperfect day of Talking Points. On the one hand, kids understood (some for the first time) that listening to somebody else could have value for them. On the other hand, many spent most of the exercise not listening to each other and simply waiting for their own turn to talk.

This doesn't mean that it was a failure. It just means it was a first step. 

I believe that if you want students to take ownership of their own learning (and listening... and opinions), then you have to allow space for them to do it in their own perfectly imperfect way. I have found that when I trust the process, I get the best results.

I am posting this to help you understand that every round of Talking Points I do is not a cornucopia of unicorns and rainbows.

On Talking Points, disidentification, meditation, and the need for a structure

One of the things I have noticed with discouraged math learners and Talking Points — or other disidentification techniques — is that students often express a kind of euphoria afterwards. “That was so much fun!” they will often say (or yell).

This is a result of the disidentification process. They are not accustomed to speaking in their own authentic voices in math class. They have become conditioned to attending math class under what Brousseau called “the didactic contract.” Under the didactic contract — the implied contract to which they have become conditioned — they are required to check their authentic self at the door, with all its attendant messiness, blurting-out, hesitant and half-formed ideas. Instead, the didactic contract demands that they conform to very narrow ideas of what a “good math student” is supposed to do: Sit down. Shut up. Pay attention. Get the right answer. Don’t ask why.

There’s no point in our denying that this is what most math students have become acculturated to. They didn’t make up these requirements themselves. Somewhere along the way, everybody encounters a math learning environment in which these are the expectations.

This situation is what the great Swiss psychoanalyst Alice Miller spoke of in her book, The Drama of the Gifted Child. In an adult-centered, authoritarian society (which most societies are), the social constructs are organized to side always with the adult/parent/teacher instead of with the child. And in fact, as Miller’s work showed, not just *instead of* the child, but at the child’s expense.

This is the general situation for discouraged learners in math classes. Many students perceive early on that their authentic selves are not welcome here. They quickly learn to wear a mask in math class and to pretend to be smart, compliant, and “mathematical” — in other words, to adopt a false persona in math class.

The problem for us as math teachers is that it is not easy to crack through this false self. Disidentification is not an easy or a straightforward process. The psyche adopts these masks as defense mechanisms — and for very good, if outdated, psychological and emotional reasons: fear of abandonment, fear of humiliation, fear of shaming, fear of annihilation.

These may be outdated, outmoded fears by the time a student reaches middle school or high school, but that does not make them any less real or active in the present moment. For a traumatized math learner, they are the most real thing in their world during 5th period.

In their own different ways, the psychologists Eugene Gendlin, A.H. Almaas, and Francine Shapiro have all posited that trauma in the past forms a kind of “stuck place” in the human mind/brain/psyche. Whenever we encounter a stimulus that “triggers” that stuck place, we “flash back” to the moment of trauma and our defense mechanisms lock into place.

This is what makes disidentification so difficult to achieve in practice. A defended psyche is not a receptive psyche. And a student may *hear* that s/he needs to adopt a growth mindset in math class, but s/he hears this message from his or her bunker, thirty feet under ground and behind several feet of concrete protective functions.

Raise periscope. Spot the threat. Lower periscope and retreat.

Evolutionary psychologists consider this fight-flight-freeze response and its replay during anxiety dreams as a most ancient form of threat rehearsal. Knowing what they know from their previous experience, the protective functions of the psyche leap into action and do their best to make sure we remain vigilant and safe from incoming threats. They perceive this to be a matter of survival, which is why they go to such great lengths to make sure we perceive it that way too whenever we step over the threshhold into math class.

So the first order of business in the process of disidentification is to establish trust and to form a safe — and sane — alliance with all learners. If math class is to become a growth mindset place for all students, then it must first be established as a safe place in which to remove our masks and to return to being our deeper, authentic, creative selves.

To make any place safe for the authentic self to come out, it helps to have a structure in place. That way, the structure can provide the psychological and emotional safety (and freedom) in which we can drop down into our authentic selves.

In all forms of mindfulness meditation, this structure consists of three things: a posture, an anchor, and a timed period.

In Zen, we sit on a black cushion in the lotus or half-lotus position (or forward on a chair with both feet flat on the floor). We place our hands on our knees or in the cosmic mudra and we face a white wall. We lower our gaze to a 45-degree angle with the floor, and we anchor our attention on our breath.

Whenever our attention wanders — and monkey mind guarantees that it will inevitably wander — we gently redirect it back to our breathing.

The Vietnamese Zen teacher Thich Nhat Hanh teaches the use of a gatha, or mindfulness verse, as an attentional aid during meditation. With each in-breath or out-breath, one thinks a line of a simple verse:

Breathing in, I calm my mind.
Breathing out, I smile.
Dwelling in the present moment,
I know that this is a wonderful moment.

Which reduces to:

In,
Out.
Present moment,
Wonderful moment.

I’ll say this about Thich Nhat Hanh: you have to be a pretty evolved being to be able to teach this kind of clarity and sanity to the very countries that launched your own into chaos.

We do all of this for the whole timed period, whether it is ten minutes or 45 or an hour. Gradually, with patience and lovingkindness, we learn how to do this for longer and longer periods, until the timed period we are working with is every day for the rest of our lives.

We do this because this is our structure.

To the uninitiated, a structure might seem to be a rigid thing, but that is a misunderstanding, and I will tell you the secret: it is actually the essence of freedom.

It gives our defense mechanisms and our wounded child ego-self-psyche something important to do while we drop down into the vulnerable place where our authentic self is kept safe — beneath all those layers of protective functions, social masks, people-pleasing, snark, and our “on-stage” personas.

The structure makes it safe for a human being to reconnect with that deeper, authentic self.

So it is natural to experience a kind of euphoria afterwards. Our culture generally doesn’t encourage us to connect with our authentic selves, so when we do, many people experience it as a kind of homecoming. Intuitively, we know that it is the source of all our greatest ideas and energy and creative fire. Finally, it is a relief to drop the masks we wear and to just be fully and authentically ourselves.

The Enlightenment poet Friedrich Schiller described this experience of flow as arising from the competing impulses toward being present and toward thinking, which operate in a kind of luminous reciprocity, with their harmonious interaction producing a third impulse which he terms the Spieltrieb (or 'play impulse'):

Irresistibly seized and attracted by the one quality, and held at a distance by the other, we find ourselves at the same time in a condition of utter rest and extreme movement, and the result is that wonderful emotion for which reason has no conception and language no name.

                       — Friedrich Schiller, Twelfth Letter on the Aesthetic Education of Man

When the mind is both fully at play and fully at rest in this way, it is at home. 

And when this experience happens in math class, students are growing and truly experiencing mathematics.

This is the sanest, healthiest, richest, most creative human state I know — and I want all of my students to experience it in my math class. Only then can they connect with the growth mindset and the mathematics that are their birthright.

But the key to unlocking that moment is through structure. And for me, in my math classes, that structure is Talking Points.