### If you were the teacher of the student who drew this below, what would be your response? What do you notice in this student's written work? What do you wonder about the student's reasoning? What would you say or ask this student?

1st Grade Math Task Prompt:

*Mario wants to cut the pizza into equal pieces and give his sister a fourth of the pizza to eat. Color the piece of pizza that Mario would give his sister.*

What if, a few minutes later, you observed a student who drew this? What do you notice and wonder in this student's reasoning? What connections do you make to the first student's response? What's a question you want to ask this student? What's your next instructional move?

A third student draws the one below. What do you notice? Wonder? What connections do you see in this response to the first two students' responses? What question would you ask to better understand this student's thinking? What's your next instructional move?

*Teaching Mathematically.*

What is it? How is this different than teaching math? Teaching students? Just plain teaching?

I consider teaching mathematically to be a mindset for teachers (particularly teachers of mathematics). It is a way of viewing and engaging in instruction where **teachers are inquiring observers and problem solvers **to determine your next instructional move**.**

Teaching mathematically means to **explore and study** your instructional practices and student thinking because you are curious about the process of teaching and learning. You **hypothesize** by doing the mathematical tasks you plan to give to students and anticipate what students may do with it.

When teaching mathematically, you **collect and analyze evidence** through observations, classroom interactions, and written student work.

You **interpret results** of your analysis to make claims about what students know and can do, which helps you decide what instructional move to make next. Through purposeful listening and questioning, you **assess the validity of your claims** by checking back with students (e.g., Is this what you were thinking?", "Did I understand correctly what you were explaining?").

Now you are ready to **publish your conclusions **to others, be it to the students as a whole (via a whole class discussion), to parents (during conferences), to our site colleagues (such as in your PLC), or to the world of your online networks of professional educators (e.g., #MTBoS).

This cycle feeds and repeats itself many, many, many times over throughout a math lesson when teaching mathematically, making teaching an intellectually challenging profession. Teaching mathematically requires teachers to rely on a mathematical knowledge base that is unique to the professional work done by teachers of mathematics (Ball, Thames, & Phelps, 2008).

I wonder:

What are the connections between teachers

**teaching mathematicall**y and students**thinking mathematically**?What are the relationships between the roles teachers play when teaching mathematically and the roles students play when thinking mathematically (see Standards for Mathematical Practice)?

How do these roles differ while also reinforce one another?

What are your thoughts? Please share them.