Standards for Mathematical Practice and Inquiry

I am on a continued quest to embed inquiry into my math lessons with fourth graders, and I am finding that I am continuing to ask, "What does inquiry look like?"  As I read our book and learn about the different types of inquiry, I don't see any that fit for math. Is it enough for me to be a better questioner, or do I need to go full on Open Inquiry?  How can I do an Open or Curricular Inquiry when there are specific skills that students need to learn in math as they progress through fourth grade? 

Obviously, there is a continuum when implementing inquiry into the classroom.  Some of what I do is much more inquiry based than others.  I used to think that all inquiry was open inquiry, with students choosing their own path for learning.  With the demands of standards (CC or otherwise) now I think that most of what I do in math is going to be curricular inquiry.  Although, that doesn't really seem to fit either.  Now I am thinking that the inquiry piece may be around how to build math THINKERS in my classroom.

My current focus from now to the end of the year in math is how to effectively integrate the Standards for Mathematical Practice into our thinking strategies for math because I think that if I make these my focus, it will change my entire approach.  It will also honor the different learning styles in my classroom and allow me to take a thinking approach to whatever type of math I am teaching.

The Standard for Mathematical Practice:

describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).   --http://www.corestandards.org/Math/Practice/

The idea of working thinking routines into math class is not a new one to me. Nor is an emphasis on discourse and writing.  But I am frustrated by the way new things seem to come down the pike, without any connection to what is already in place.  With a continued concern that kids cannot transfer their knowledge from content to content, I am interested in streamlining what I am doing in terms of setting expectations for thinking and understanding in fourth grade.

My plan is to lead students in an inquiry lesson where they pull from their schema about the thinking strategies and connect the SMP to each.  We are going to do one (Questioning) together as a class and then students will do the Chalk Talk routine to connect them.

How does this all connect to inquiry?  I am thinking that if I can do a better job of setting the expectation that, "These are the ways you will be expected to think in math," then that will lead us to much richer, inquiry-based math learning.