cheesemonkey wonders

cheesemonkey wonders

Friday, April 6, 2018

HERESY WARNING: Breaking through on quadratic function analysis & graph sketching

Over the last two days, I've had a few important breakthroughs with discouraged Algebra 1 learners about quadratic functions and their graphs. I wanted to document this for myself before I have a chance to forget about it for next year.

If you have rigid beliefs about the only ways for students to approach quadratic functions, analysis, and graphs, then this post is definitely not for you.

Consider yourselves warned.

My most discouraged Algebra 1 learners are extremely gifted kids, but this year's crop are definitely dreamers, not stalkers. They need to really marinate in something for a long time before it takes root in their minds. They are factoring warriors, but the quadratic formula and complicated answers can be really daunting for them.

The connection I have wanted all of my students to make between quadratic functions and their graphs is this: even when a quadratic doesn't have neat, simple, integer answers, it still has a number of neat, simple aspects that they can grab hold of. There will always be an axis of symmetry. There will always be a vertex.

What I have discovered — or rather, what they have been teaching me all this week — is that if you approach a quadratic function from the understanding that you already know how to find the AOS and the vertex, you can make a TON of important discoveries and understandings about its graph and the many properties of the function that are important to understand.

What they taught me today is that they know how to use the axis of symmetry formula like a key in a lock. They can identify a, b, and c in a quadratic, and they understand that they can find and graph the AOS equation quickly and fearlessly. Then they can plug in the value they found for the axis of symmetry to identify the vertex of the parabola.

From there, it is a simple matter to find some more points for your sketch and to identify their mirror reflections across the axis of symmetry.

The beauty of their method is that it makes it easy for them to develop a meaningful conjecture about whether or not a quadratic function has any zeros.

If the parabola is floating above the x-axis, then they can use the QF to confirm their hunch that there are no real-number zeros to the function. Likewise if the parabola is submerged below the x-axis.

I love that they naturally figured out today what it means for the vertex to be a maximum or a minimum.

And I especially love the fact that they made these discoveries themselves.

They still don't understand how to complete the square or use the quadratic formula to blast through problem after problem to find complicated zeros of non-trivial quadratics.

But this feels less important to me than the fact that they have made important connections and developed their own methods for investigating quadratic functions. And it has been an important reminder to me to design learning experiences that empower them to make these connections and discoveries for themselves.

Sunday, January 28, 2018

The Centrality of the Cross-Racial Therapeutic Alliance in My Classroom

As part of our school's five-year program of equity transformation, we the staff engage in monthly two-hour seminars focused on issues of equity and diversity. As someone who is leading (and learning to lead) equity transformation in my very large school, I live on both sides of the fence in this process. As a peer, I am often a witness to frustration and discouragement that naturally arise in an inherently vexed and vexing process, since we are only in year two of the process. And as someone on the leadership team, I am often tasked with encouraging greater engagement in a program whose theory of action is I sometimes have issues of practice with.

Nevertheless, I wanted to explore my own understanding of an important part of this month's reading included an article titled, "Racial Microaggressions in Everyday Life: Implications for Clinical Practice," by Derald Wing Sue et al (May—June 2007, American Psychologist). 

Most of my colleagues' responses to this piece have centered on the numbingly detailed taxonomies the authors present about forms of microaggressions, microassaults, and microinvalidations. This part of the article felt deeply Aristotelian to me — busily categorizing, classifying, analyzing, exemplifying, and naming these forms of everyday racist aggressions that occur every day everywhere in our country.

What has struck me most in this article — and what has stayed with me — was the discussion toward the end of the piece about the centrality of the therapeutic alliance within cross-racial dyads. This hit home for me because it is what I experience as the most central part of my own work in my classroom with my students of color, so I noticed myself sitting up straighter and taking much better notes when I was reading this section.

A lot of research on the effectiveness (or ineffectiveness) of the therapeutic process has zeroed in on the therapeutic alliance as the most essential determinant of success in personal transformation. In therapy, this means that the client must perceive the therapist himself or herself as being both deeply trustworthy and expert in the process of transformation. It seems to me that this same relationship exists within the classroom. The bond I have to form with my students requires that they perceive me as both worthy of their trust and expert in the process of transformation.

In addition to the client's perception of the therapist, the client must also have a very specific perception of him- or herself within the therapeutic context — namely, they must perceive that they themselves are being deeply understood and received in a relationship of unconditional positive regard. This too is a critical part of my own culturally responsive pedagogy. My students must perceive themselves as being welcomed to "come as you are" in my classroom because I know that this is the only thing that will make them feel safe enough to become vulnerable enough to engage in the transformative mathematical identity work that we do in my classes.

A huge part of my own practice of trustworthiness with my students  is about being open to accepting responsibility for harms that I cause (or perpetuate) and for taking corrective actions to repair the harms and clean up my side of the street in our relationships. This becomes even more important as a prerequisite for cultural competence, because I am sure there are a zillion unconscious ways big and small in which I, as a White teacher microassault, microinsult, or microinvalidate my students of color while at the same time using every tool in my power to love and support them. Being in community is like being in any relationship. You can never fully avoid running into conflict, but you can learn better ways to repair the local harms that arise so that the underlying fabric of the relationship remains intact for the benefit of all parties.

Because most helping professionals, including teachers, experience themselves as fair and decent people who are incapable of causing deliberate racist harm toward our students of color, it can be difficult to see microaggressions as they are happening because they are so often invisible to authority figures who are accustomed to swimming in a soup of white privilege. Yet I must continue to practice noticing my own white racial identity in order to develop this capacity in my conscious mind and social practice.

For me, the most urgent idea in this article was the fact that when racial microaggressions continue to occur within a cross-racial therapeutic context without being noticed, named, and addressed, clients of color are at risk of not continuing in their own work of inner development. To put it in the context of my classroom, if students perceive that I perceive myself to have no responsibility for noticing, naming, and addressing racial injustices at the micro- level within our relationship, then they are more likely to become discouraged and give up.

I don't know where all this thinking is going to take me. I only know that framing the cross-racial teaching and learning relationship as a therapeutic alliance feels transformative to me. And that means I need to keep investigating my role in it.

Sunday, September 17, 2017

Conversation With Grace: Why Strive For Diversity?

Ever since I read Grace's post, I've been struggling.

I have nothing but bad, inadequate answers to this question. And since its premise lies at the heart of my every day's work, I have felt stuck as I struggle to find a better answer.

I still have only terrible answers, but I need to get unstuck so I can get going again. So I'm writing this post as a chance to stick a pin in the best bad answers I have right now.

1. tikkun olam
Tikkun olam is the central assignment in the spiritual curriculum I grew up in. It is the requirement that we work actively to heal the world. When I was a child in the 60s, our rabbi was an active figure in the civil rights movement — a friend and colleague of Dr. King's, one of the Freedom Riders, and "the most arrested rabbi in America." As Dogen Zenji, the founder of the Zen lineage said, "When you walk in the mist, you get wet." Spending day after day, week after week in this community, I walked in the mist. I got wet. I came to believe in the power of tikkun olam to make America a more just place, even if it was still massively imperfect. Tikkun olam also taught me that if we as a society have the possibility of providing quality education and access to power for some, then we need to make sure that we make that possibility a reality. Our society and our world are a mess right now, so there's no time to lose. And the only way to heal the world is to actively work toward healing the world. I have to accept that I may be doing it wrong or badly, but in making positive effort for the good each day, I am doing the best that I can. And the more people I reach, the better. Because I never know who will trigger...

2. the hundredth monkey principle
In the 1960s and 70s, the hundredth monkey principle became an archetypal myth and story for me about how critical mass seems to come about in human society. The story has been used and recycled in many different contexts, but what strikes me as important is this: the story provides a useful metaphor for me about what happens as we share good ideas and carry them with us. For me, the big idea is that we can have no idea where or with whom the tipping point or critical mass will happen. So sharing widely — and diversely — is important to me because I have no idea who will be that "hundredth monkey." Identifying the hundredth monkey is above my pay grade. My job is to share best practices and wisdom so that it reaches the hundredth monkey and we can achieve some degree of tikkun olam sooner rather than later. The more people I reach, the better.

3. diversifying the power structure
I was grateful that Julie Wright shared this clip of Michelle Obama sharing her beliefs about the power of diversifying the power structure after watching the film Hidden Figures. Michelle and I were at Princeton at the same time, and we both benefited enormously from its diversity initiatives. I was in only the tenth class of women at Princeton, and Michelle was two years behind me. The education we received there was transformative, but it was also transformative to live in community with others from such dramatically differing backgrounds from around the world. It was my first experience of gifted education, and it changed me. It changed a lot of us. Several years later, I was privileged to be a part of a conversation at a conference in which Henry Louis Gates, Jr. framed the need to diversify the power structure and the power elite if we want true transformation, and I believe that for better or for worse, he is right. If we want to transform our society, we need to transform its power structure, and for that, we need to diversify our classrooms. As I've gotten older, I've gotten less apologetic for my elitist tactics here. I have less time to waste, and I want a better quality of leaders across the board in our pipelines.

4. Yes, it is better for me.
This is the argument I feel most ashamed of, but I have to be ruthlessly honest about my own motives. I hate it when I have to admit that I am ruled by enlightened self-interest, but it's a reality in economics and it's often a reality of my own life. It has nothing to do with pity for others and everything to do with what makes my own experience in the world I live in better. I love exploring the world around me, and that means I love experiencing viewpoints and perspectives that are different from my own. In my previous career in high tech, I experienced how blind spots take over when everybody at the table looks the same. In education, I see how the wealthy and powerful push idiotic agendas when they only listen to other wealthy and powerful people.

So yes, I work for greater anti-racist diversity in my educational context because it is better and healthier and saner for me.

I have a strong suspicion it is also better for others, but by the principles of tikkun olam, I am restricted to cleaning things up on my own side of the street.

I would love for us to have ever more diverse TMCs, but I am also bound to respect the differing boundaries, needs, and wants of the people who would make those things more "diverse" for people like me. I am bound to respect that for those who spend their school year isolated within an oppressively dominant culture, it may not feel optimal to them to spend their summer resources attending a PD conference that is still largely populated by teachers from the dominant cultures. That puts the onus on me and others like me to strive to make TMC a worthwhile and valuable choice for them to make. But it's critical to recognize and respect that those who would make a TMC "more diverse" place for me have their own needs, wants, and priorities. I need to make sure I am not trying to manipulate others into serving my own needs, wants and priorities.

I cannot bear to #pushsend, but I also can't bear to feel stuck any more.

I want to get back to giving my wild and imperfectly diverse classes the best educational opportunities I can give them, so I'm going to #pushsend anyway.  God help me.

Thursday, August 24, 2017

Why does this feel so dangerous...?

Maybe it's just the cold medicine speaking (I am battling a wicked cold), but today in 6th block Algebra 1, as we were working on Exeter problems, I ditched my usual, "safe" piece of lecture and went went for the jugular.

I wanted to give them the real stuff — not the party line position.

I advocated for radical, student-centered sense-making.


Saturday, July 22, 2017

"You are what you seek," the wise one said.

Once upon a time, some time late in 2011, there were some lonely, kooky, determined math teachers trying to get better.

They searched blogs and joined Twitter in their quests, and eventually they found some like-minded spirits on the internets who were also questing.

In 2012, forty of us decided to meet in person in St. Louis and hold our own conference. Nobody outside that first group (besides Fawn, who hosted #TwitterJealousyCamp) knew or much cared about what we were doing. We were doing it because we wanted to do it. Period.

FUN FACT: Out of forty attendees at the first TMC, I was the one and only attendee from California. In fact, we had more attendees from Mississippi than from California.

It wasn't perfect, but it was real — and that kindled a spark. What made it magical was the fact that people showed up and brought their A game. I learned something amazing from every single person at that conference.

So if you attending TMC for the first time this year, please temper your freaking out with the knowledge that we started this thing because we were looking for YOU. We are STILL looking for you.

Before TMC, I always think of one of my favorite quotes from the great Jungian psychoanalyst and storyteller Clarissa Pinkola Estes:
  Even though there are negative aspects to it, the wild psyche can endure exile. It makes us yearn that much more to free our own true nature and causes us to long for a culture that goes with it. Even this yearning, this longing makes a person go on. It makes a [person] go on looking, and if she cannot find the culture that encourages her, then she usually decides to construct it herself. And that is good, for if she builds it, others who have been looking for a long time will mysteriously arrive one day enthusiastically proclaiming that they have been looking for this all along.

Friday, July 21, 2017

#TMC17 MORNING SESSION OVERVIEW: Differentiating CCSS Algebra 1 — from drab to fab using Exeter Math 1 & Exploratory Talk

How can we use a problem-based throughline such as Exeter to transform a lifeless but mandated Algebra 1 course into a rich, differentiated experience of mathematical sense-making for a wide range of students?

This morning session will be a master class in differentiating a generic Algebra 1 course using problem-based learning, exploratory talk, PCMI-style differentiation, and deliberate practice in its appropriate place together with metacognitive self-monitoring.

Over the course of our time together, we will move back and forth between the perspective of learners and the perspective of teachers. During reflective "master class" segments, we will explore the theories, techniques, and practical aspects of rearchitecting Algebra 1. During immersive “math-doingsegments, we will do selected sequences of problem-based mathematics together in groups so we can experience different approaches to concept development, cultivating habits of mind, building norms through math content, and engaging the whole student through experiential problems. Immersive segments will be interwoven with reflective, “master class” segments in which we will analyze the theories, techniques, and ideas we're exploring.

Here is the 30,000-foot overview of the topics we'll be digging into over our three days.


Brief review of the How People Learn (HPL) learning cycle so that we will have a shared vocabulary for our work together. EQ: What research research informs these ideas about teaching and learning with understanding?
Practical strategies and tips for organizing and managing your own teaching and learning of Exeter sequences to support your work with students.  EQ: This feels overwhelming— how can I set myself up for success?
Why and how Algebra 1 must be reimagined as a course in advanced proportional reasoning. 
Through experiential "doing" segments and reflective discussions, we'll explore some of the ways in which the anchor problems and supporting problems within the Exeter sequences encourage students to get inside the problems in a state of flow rather than killing time filling in charts with mindless data-gathering.  EQ: How do the Exeter problems cultivate a stance of shareable curiosity?

Strategies for integrating Talking Points as a focused technique for developing collaborative speaking and listening skills.
Structuring your room and tasks to support the needs of both katamari and speed demons. EQ: How can I create an environment that consistently values all student ideas and thinking?

Anchoring students' mathematical thinking in the body, in the present moment, and in the value of their own existing knowledge and understanding.
How and where to integrate practice activities in ways that support student agency, dignity, and understanding.

Friday, July 7, 2017

Things That Work #1: Regular Vocab Quizzes in Geometry

One of the things that worked incredibly well last year—and which I want to extend this year—is regular vocab quizzes in Geometry.

Vocabulary is the gating factor for success in a problem-based, student-centered Geometry class. If you can't talk about geometry, you can't collaborate about geometry.

I learned the value of extremely routine-looking vocabulary quizzes when I taught 8th grade English with Alec MacKenzie, Linda Grady, and Kelly Starnes. At the beginning of the school year, the copy room delivered us each a giant stack of very basic matching quizzes: numbered terms in the left-hand column, lettered definitions on the right. Each student got a vocabulary workbook at the beginning of the school year. Every week we assigned a new chapter/list. Every week we gave a matching quiz. And then we would trade and grade them.

At some level, I recognize that this sounds stultifying. But at another level, it was incredibly empowering for the students. Everybody understood exactly what was being asked and expected. And everybody saw it as an opportunity to earn free points. Students gave each other encouraging written comments and cheered each other on. They saw their scores as information—not as judgment. They used what they knew to make flash cards or Quizlet stacks. They quizzed each other. They helped each other.

And nobody ever complained about the regularly scheduled vocab quiz. It was a ritual of our course.

Vocab quiz for initial unit on circles
In my first few years of teaching Geometry, I have noticed that the kids who make the effort to integrate and use the vocabulary and specialized terms tend to succeed. And the kids who don't use the language of geometry suffer. So I decided to use what I know to raise the number of kids who know and use the vocabulary by instituting regular vocabulary quizzes for the relevant lessons or chapters as we go.

Many of my discouraged math learners sprang to life when I assigned this task. They pulled out flash cards, folded sheets of binder paper in half lengthwise, and started organizing the information they wanted to integrate. In most of my classes, I noticed that the highest-status math students often seemed to get stuck while the weaker students knew EXACTLY where to start and what to do.

It was a revelation.

It also ensured that everybody spent a little quality time on the focus task of preparing for the vocab quiz on Thursday or Friday. And this, in turn, meant that everybody was a little more ready to use the correct and appropriate mathematical vocabulary in our work. They noticed more because the owned more.

Because these were "for a grade," kids put their shoulder into it. My colleagues in other departments commented about my students taking two or three available minutes during passing period to quiz each other.  It gave them hope.

Now I want to create a full set of vocab quizzes for my whole year. 

A few implementation notes:
  • I collect and shred/recycle all of the quizzes after I enter their scores so I can reuse the same quizzes from year to year. If I don't have your quiz, you can't get a score. I am strict about this.
  • Every new vocabulary term does not have to get quizzed, but lessons or units where there is a huge vocabulary burden that gets front-loaded deserves its own vocab quiz. I have been surprised to discover how many lessons are more vocabulary-intensive/language-intensive than I had realized.
  • Correct use of technical language is self-reinforcing. Once I introduce a new term, I mercilessly ask kids to remind each other of the definitions for 15 seconds in their table groups. Getting one kid to call out the correct definition to the whole class is not the point here. Getting 36 kids to all speak the definitions or the terms in their table groups is.
UPDATE: D'OH! I can't believe I forgot the most important implementation note I wanted to remind myself about!!!
  • There should be many more definitions in your right-hand list than there are terms in your left-hand list. Also definitions can be re-used. This way there isn't a zero-sum outcome if someone misses an answer.