I am trying to make a commitment to myself to "write rough" here more often. My hyper-analytic personality leads me to think and rethink things so much that I rarely get to a mental space in which I feel like I can write with any clarity. So, this might be rough.

I attended the Creating Balance conference up in San Francisco last weekend, where Rochelle Gutierrez gave a talk about "Teaching Mathematics as a Subversive Activity." Her talk hit close to home for me but, even given it's closeness, still has left me thinking and rethinking her words this past week.

I appreciated many things about her talk, but mostly the critical way in which she addressed the sort of taken-for-granted discourses, or structures, in mathematics education that politicize the issue. Things like success, proficiency, achievement, and even what counts as mathematics largely have a singular meaning. As she writes,
"What counts as knowledge, how we come to 'know' things, and who is privileged in the process are all part and parcel of issues of power."
Mathematics education deals primarily with the dominant view of mathematics, the "discipline." With such a singular view we fail to recognize that, as Dr. Gutierrez points out, "mathematics needs people as much as people need mathematics." She writes,
"Most often, the goal in mathematics teaching is to try to get the student to become a legitimate participant in the community of mathematicians, thereby subsuming their identity within the currently sanctioned way of communicating in the field."
"Yet, when students offer a different view, they are seen as having deficient, underdeveloped, or misconstrued understandings of mathematics."
Dr. Gutierrez describes this as the "deficit model" which I see as very similar to Friere's "banking model." Attending to dominant mathematics can sometimes mean losing oneself at the expense of adopting another's way of knowing. As Dr. Gutierrez reminded us during her speech, teachers of mathematics are "identity workers." When we define mathematics narrowly and we impose that mathematics, we fail to recognize the mathematics of the individual, the student, the child. These are issues of identity (is my thinking mathematical?), of equity (who gets to participate in mathematics?), and of power (who benefits as a result of this?).
The point where the two axes intersect is a space Dr. Gutierrez has named "Nepantla." Borrowed from the work of Gloria Anzaldua, Nepantla is an Aztec term that refers to "el lugar no lugar" (neither here nor there). It was best described, I think, by Dr. Gutierrez to simultaneously mean "both" and "neither." I think it succinctly captures the paradox and tension I have felt in my research on student agency and identity. The discourses and structures in society require that we attend to dominant mathematics (access and achievement), yet I know there is an alternative in which moving away from dominant mathematics allows us to attend to the mathematics of students/people (identity and power). We can't do both but we also can't do neither. So, as teachers, we live in Nepantla...we live in tension.

It is from this space of Nepantla that new options, new knowledge emerge. To retreat to safety, to settle in the current ways of doing things because no clear alternative is present, is to choose to live in "desconocimiento" (a distancing, ignorant space in which we refuse tension). The alternative is to choose tension, to live in the messy space of Nepantla. Dr. Gutierrez describes curriculum as "both a mirror and a window." A mirror because it allows the student to recognize oneself in the work they do, but a window because it also allows them a new perspective on the world. I don't know that it's attainable, but I think we must choose to pursue the unrealizable philosophical ideal and live in messiness because to settle is to distance ourselves from knowing mathematics with our students.
I love the Khan Academy videos. I love them because they have opened up a discussion about education, learning, theory, and bringing all of it together in meaningful ways to support students in their mathematical development. Relatively new to the education scene, Khan Academy receives a lot of criticism. In the most recent critique of Khan Academy, we have been amused by the "Mystery Teacher Theater 3000" videos.

First, I think that Khan Academy gets a bit of a bad wrap. I'm pretty sure it was not Sal Khan's intent to "revolutionize" or "reinvent" education by posting his videos online. In my opinion, a public misconception about mathematics, learning, and education has allowed KA to rise to it's current position. The public views mathematics as a body of factual information, one that is passed from the "knower" to the "student." Learning, then, is perceived as memorizing/understanding factual information. This public perception applies to much of education in general, but especially to mathematics education. 

Recently, the MTT3K video exploited some mistakes in Khan's explanation of multiplying positive and negative numbers. Among other criticisms, the makers of the video point out that Khan mistakenly refers to the "transitive property" and explains "-4 x 3" as "negative fours times itself three times." In addition, the video points out some questionable pedagogical decisions by Khan which might be confusing or difficult for students first learning the topic and that fail to explain why the multiplication facts are true.

I welcome the dialogue that is happening surrounding these videos, but want to take the discussion of why Khan Academy is an ineffective learning tool one step further. In doing so, I want to reference Constance Kamii and her application of Piagetian principles to teaching:

"Piaget's theory of memory is very different from the empiricist belief that 'facts' are 'stored' and 'retrieved.' According to Piaget, a fact is 'read' differently from reality by children at different levels of development because each child interprets it by assimilating it into the knowledge he has already constructed."

In this next quote, she refers to facts about addition. I think the same could also easily be said for multiplication:

"In Piaget's theory, there is no such thing as an 'addition fact.' A fact is empirically observable. Physical and social knowledge involves facts but not logico-mathematical knowledge. The fact that a ball bounces when it is dropped is observable (physical knowledge). The fact that a ball is not appreciated in the living room is also observable (social knowledge). But logico-mathematical knowledge consists of relationships, which are not observable. Although four balls are observable, the 'four-ness' is not. When we add 4 to 2, we are putting into an additive relationship two numerical quantities that each of us constructed, by reflective abstraction; 4 +2 equals 6 is a relationship, not a fact."

At first, these quotes seem hard to swallow. However, Piaget (and others) have conducted many experiments to show that even children who "knew" facts about addition lacked the ability to perform on different tasks involving those same operations.

We construct in a way that is personal to us and relative to our current ways of understanding. My point in all of this is that, video or no video, you can't make someone learn. The 'knowing' that is so prized in education comes from every person's natural ability to think and from the human tendency to maintain internal/cognitive equilibrium. My suggestion is that we don't tell students what they should know or how they should think. Give them a task that pushes on their way of knowing, let them do mathematics, and watch and listen closely as they sort things out together.
I am nearing the end of my first year in grad school. Even more than I anticipated, blogging and getting feedback from readers has been exceptionally helpful in pinpointing the things in mathematics education that really matter to me the most and in finalizing my action research proposal. With that said, I wanted to post my preliminary research proposal in case anyone is interested in reading it. It is rather long and still relatively "un-polished" but I owe much of it's creation to readers who regularly push my thinking. Thanks!

How does a mathematics classroom centered in habits of mind support students in realizing mathematical agency?

What does it mean to "educate?" Here is dictionary.com's definition of education:
The next section was alarming:
I'm just (very) concerned with the perception that education is all about the transfer of "facts." I know this isn't a new argument, but I can't help but feel this is a pretty critical time in education (particularly for math). National and state budgets are in bad shape. People perceive math as a fact-based discipline. Computers provide fact-based instruction effectively. I find all of this really alarming.

There was a great post by Grant Wiggins recently in which he suggests:

"I propose that for the sake of better results we need to turn conventional wisdom on it is head:  let’s see what results if we think of action, not knowledge, as the essence of an education; let’s see what results from thinking of future ability, not knowledge of the past, as the core; let’s see what follows, therefore, from thinking of content knowledge as neither the aim of curriculum nor the key building blocks of it but as the offshoot of learning to do things now and for the future."

And another section from the same post in which he references Ralph Tyler:

"According to Tyler, the general aim is "to bring about significant changes in students’ patterns of behavior.” In other words, though we often lose sight of this basic fact, the point of learning is not just to know things but to be a different person – more mature, more wise, more self-disciplined, more effective, and more productive in the broadest sense."

To me, these quotes speak to what it means to "educate." I think we would be well suited to redefine math education as one that nurtures and cultivates habits of mind. These are the habits that influence future action, that inspire innovation, and that, in my opinion, are what it means to be "educated." More to come...