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."

 
 
You always hear people say, "kids don't like math!" Correction...kids don't like feeling dumb. People don't like feeling dumb. Feeling dumb comes from being told you're "wrong" over and over. Math education effectively does this better than just about any other subject in school. It's no wonder that people are sending out tweets like this:
Things get even worse when we start throwing grades in the mix. Now, all of a sudden, not only are we telling kids they're wrong...we're punishing them for it. It shouldn't be surprising that students are reluctant to take risks, persist with difficult problems, and trust their own thinking; they don't want to get it wrong. Today, I saw this tweet:
To which, I replied:
On second thought, I should have written: "It CAN'T exist independent of a specific group of kids." I mean, of course we know it can, and it does, but isn't this putting the cart before the horse?!? When your curriculum is decided in advance, you've already told students what the measure for "knowing" is and if they don't meet that then something is wrong with them. I've written about other curriculum/lesson structures that respond more directly to students but my concern goes beyond that.

I have had the pleasure of "mentoring" (which is a total misnomer...we really just learn from each other) one of our new teachers this year. She's amazing. Yesterday we were planning a probability unit together and we were trying to figure out an answer to the unit question that would drive our investigation of probability. In the process, her and I were collaborating, conjecturing and testing, investigating smaller problems, drawing diagrams, and listening closely to each other. Honestly, I don't really care if, as a class, we are successful in calculating an exact answer to this problem. I want students to come up with an answer that makes sense to them, that responds to their ways of knowing, and that is reflective of their deepening understanding of chance and probability. To quote a brilliant mentor of mine, "I guess my point is, 'solving' the unit problem will certainly be in the discussion but we'll be 'successful' moreso because students invented the solution, rather than being told." Mostly, I just want the students to have the same experience we had; the experience of playing with and doing mathematics.

I know there are "realities" in a lot of schools that make this difficult. Benchmarks, AYP, API, etc, etc. There are all sorts of measures of success and progress out there (most of which, I would argue, are false indicators of "learning"). Well, here's my vote for success/progress:
I want people to know that we are all mathematical in our thinking...maybe just not in the ways that school has defined mathematics. I want fewer people to hate math.