K-12 teachers shape the public view of mathematics.  They are the first representatives of the math community.  If we want the public and our future students to understand what math is really like, we need teachers who have acquired mathematical practices: habits, values, and characteristics that enable them to independently generate and access mathematical knowledge, reason and communicate mathematically, and create authentic mathematical experiences for their students. Thus, in teaching preservice teachers, we feel it imperative that we focus our teaching on helping them develop mathematical practices, some of which are found in the SLOs, such as: making sense of problems and persevering in solving them (e.g. SLO 2 & SLO 10), generating and utilizing mathematical representations (e.g. SLO 6 & SLO 8), reasoning mathematically and constructing viable arguments (e.g. SLO 1 & SLO 5), communicating mathematical ideas through the precise use of language and symbols (e.g. SLO 5), and understanding and critiquing the reasoning of others (e.g. SLO 1).

Studying our students’ and our own math practices has empowered us to support our students in developing these practices. As we describe in Chapter 47 of the GeT book, we study practices through a cycle of inquiry that involves: identifying evidence of math practices in student, creating & implementing tasks to engage them in those practice, and assessing their practices. This allows us to identify micro-practices: fine-grained skills that contribute to the development of a practice. Awareness of micro-practices enables us to provide explanations, orchestrate discussions, create tasks and assessments, and provide feedback that helps our students acquire math practices.

For example, by studying the practice of reasoning mathematically and constructing viable arguments (SLO 1) we identified a number of micro-practices that we can intentionally discuss, help students develop, and assess.  These include that mathematically proficient students:

1. Justify claims and solutions through a convincing explanation as to why it is correct, rather than simply telling how they solved it. 
2. Attend to the scope of a claim by providing examples and counterexamples, when they are sufficient, and broadly applicable arguments, when examples are insufficient. 
3. Ensure their arguments are complete by identifying and attending to the burden of proof.
4. Build their case by making valid and relevant claims. 
5. Provide detail by supporting all of their claims. 
6. Make valid deductions from appropriate axioms, definitions, and theorems to support their claims. 
7. Logically structure their arguments in valid ways, appropriately using direct proof, proof by contradiction, proof by contraposition, or proof by induction techniques. 

Knowing these, we help students develop them through tasks and assignments, discuss these outright, and use them in rubrics for grading and feedback.

The inquiry cycle has become a part of our ongoing work as teachers.  It gives us a way of exploring other math practices and support students in developing them.  The benefit of identifying micro-practices goes beyond our geometry classes.  It allows us to support student development across the curriculum and provide early mathematical experiences that contribute to our students’ development of productive mathematical practices. 

We encourage you to take a deeper look at how you implement mathematical practices in your own courses and begin your own study of your and your students’ math practices. For additional information on our cyclic approach and in-depth descriptions of aforementioned micro-practices, see our paper.