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.