This post addresses expectations. It also addresses many of my skepticisms about things I've seen in the How to Learn Math course and in reading Carol Dweck's Mindset. I think it comes down to expectations: expectations of ourselves and expectations of students.
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So one of the challenges that comes in teaching math is the wide range of students' skills and retention from previous courses (hell, even retention from the current course). Our school has done a lot of work with differentiation, assessments, learning targets, etc. One of the ideas in a lot of the differentiation literature is that students should not be assessed the same way. The standards should be the same, but not necessarily the level of the tasks. I think myself and most (maybe all?) of my colleagues are extremely skeptical of this idea.
Yet, the growth mindset comes from helping students grow rather than judging them. It seems to suggest further evidence against using grades. But, if you're in a situation where you have to give a grade, is it fair to give the same grade to two people when one performs way higher than the other. One person is exceeding a standard, one can do it consistently, and one only has a really basic, incomplete understanding of it. It's one thing to say they can meet a standard. It's another for them to have understood it at a level sufficient to be able to use it for future mathematics. I know "Fair Isn't Equal," but when you have external forces (parents and colleges) looking at these grade marks, the reality is it still is important, regardless of whether or not it should be.
But then, if you expect students to still do their work without any grade to work for, then it has to be "engaging." Even if they are getting grades, the work still has to be engaging, anyway. Any good teacher understands the importance of engagement. However, I think there is a great danger in engagement for the sake of engagement. I don't believe you can sacrifice content. If you do, then I think you're mortgaging their success in future courses so that they don't feel like a failure now.
So I think a lot of the skepticism boils down to this question of what is important: "Is it what they learn, or how they feel about it?"
Because regardless of whether or not it's compatible with students' interests, they have to learn how to write an equation of a line correctly. Plus, anytime you try to engage the student with a context, you run the risk of dis-engaging them further because now it's a word problem.
And even if you "explain what you're doing" or "correct and explain the mistakes you made," I still think it's very important to be to actually do the problem correctly. I agree with the process being more important, but getting the correct answer (and Attending to Precision) is still part of the process. Part of the constructive dialogue and feedback probably still needs to be "This still isn't correct, and you still need to work on it more." I'm not sure that you can say someone is proficient at something if they still are not getting a correct answer.
I also find it a little perplexing the dueling messages we get. We're expected to align our curriculum to Common Core, which assumes that everyone learns the same things the same year (and the fact that whether or not they can demonstrate this on a test will determine teachers' livelihoods.) However, we're also supposed to be using differentiated instruction, which operates on the premise that everyone learns differently and at their own pace.
