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...
I have been writing a bit about the problem with "right" answers in the mathematics classroom lately. I think it is a big concern. Upon further reflection, I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:

"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."

There is some great reading about how this relates to "mathematics for all" and teaching for social justice. Building this community is extremely difficult. Students have developed expectations about what learning and teaching mathematics "should" look like. As a result, if we are to promote this type of student discourse it becomes necessary to renegotiate the "didactic contract" (Brousseau, 1986). This "contract" both explicitly and implicitly outlines the role of both teacher and student, the expectation for classroom discourse, and, as a result, the locus of authority.

This type of discourse simply isn't going to take place if "mathematics" is practiced as "solving" a bunch of related problems (read: 1-30 odd). Instead, find the big idea, pick one rich question to lead an investigation of that idea, and then let the students sort it out. You are likely to see students doing mathematics. To return to Schoenfeld,

"This <is> their mathematics. They <have> ownership of it, not only in the motivational sense, but in the deep epistemological sense that characterizes the true mathematical knowing and understanding possessed by mathematicians."

I just finished reading three selections from Alan Schoenfeld's "Mathematical Thinking and Problem Solving." The book is incredible and I would highly recommend it. It features eight different papers that are each counterbalanced by a written response or discussion. This may be a lengthy post but I wanted to record my thoughts here for later reference. In an effort to promote conversation about these, I would encourage you to find at least one quote that is interesting to you and post a thought or question in the comments. 

Research and Reforming Mathematics Education (Schwartz)

1. "If the important idea that one devises is to find its way into the way mathematics is taught, learned, and made in schools, then it is important that it not appear to threaten current practice." (p. 3)
2. "It should appear to augment, rather than replace." (p. 4)

"However, let it be clearly stated and recognized, that any curricular development that takes place without the continuing central and focal involvement of teachers is almost certain to fall short of its potential."

"Less visible, but equally if not more important, is the role of the social network in legitimizing the adoption of new habits of mind. It is difficult for students well into a school career in which mathematics has been an endless series of incompletely understood calculation and manipulation ceremonies to shift gears and to exercise in class their curiosity and inventiveness. It is difficult for teachers and principles, who will be held accountable to superintendents and school boards to imagine mathematics classes in which mathematics is 'discovered' rather than 'covered.' It is difficult for school boards to imagine that the mathematics they learned and the way they learned it is possible not universal or eternal in its importance. All of these groups need support as they develop new 'habits of mind.'" (p. 6)

Doing and Teaching Mathematics (Schoenfeld)

1. "In turn, our classrooms are the primary source of mathematical experiences (as they perceive them) for our students, the experiential base from which they abstract their sense of what mathematics is all about." (p. 53)
2. "The activities in our mathematics classrooms can and must reflect and foster the understandings that we want students to develop with and about mathematics." (p. 60)
3. "When mathematics is taught as received knowledge rather than as something that (a) should fit together meaningfully, and (b) should be shared, students neither try to use it for sense-making nor develop a means of communicating with it." (p. 57)

1. "The implicit but widespread presumption in the mathematical community is that an extensive background is required before one can do mathematics." (p. 65)
2. "I work to make my problem-solving courses 'microcosms of selected aspects of mathematical practice and culture,' in that the classroom practices reflect (some of) the values of the mathematical community at large." (p. 66)
3. "Mathematics is the science of patterns, and relevant mathematical activities - looking to perceive structure, seeing connections, capturing patterns symbolically, conjecturing and proving, and abstracting and generalizing - all are valued." (p. 68)

Classroom Instruction That Fosters Mathematical Thinking and Problem Solving (Romberg)

1. "What is constructed by an individual depends to some extent on what is brought to the situation, where the current activity fits in a sequence leading toward a goal, and how it relates to mathematical knowledge." (p. 300)
2. "The 'intended' curriculum can only include our best guesses about what will both interest students and lead all toward development of mathematical power. At the same time, the 'actual' curriculum depends on teacher choice, and the 'achieved' curriculum depends on each student's interest and prior knowledge." (p. 300)
3. "Nussbaum and Novick suggest a three-part instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)

1. "Classrooms should be places where interesting problems are explored using important mathematical ideas…This vision sees students studying much of the same mathematics currently taught, but with quite a different emphasis."
2. Romberg calls for a problem-based approach centered around his five principles of curriculum design (conceptual domains should be specified, domains should be segmented into 2-4 week units, students are exposed to domains as they arise naturally in problem situations, activities are related to how students process information, and curriculum units should always be considered problematic).