You're using standards to develop learning goals for your students. You have found a mathematics task, such as the Ticket Booth from Illustrative Mathematics for example, that promotes problem solving and reasoning. Now you give the task to your students, right? Before doing that, ask yourself, "What's my next move?"

To answer that question, I am finding the 5 Practices to be helpful, particularly when supporting teachers' planning. I've begun collecting tasks that promote reasoning and problem solving (see anticipatingtheirideas.blog) where I've started anticipating how students might approach the tasks. Feel free to visit that site and provide your predictions of student thinking.

When using the 5 Practices framework for orchestrating mathematical discussions, there are two timeframes these practices occur.

- The last four practices (monitoring, selecting, sequencing and connecting) occur during a live classroom lesson
- The first practice, anticipating, occurs
*prior*to the lesson during a planning session

Anticipating how students will attack a task is the action of this practice. Some general questions I consider about student thinking on tasks that encourage multiple strategies, are:

- What are some typical strategies students will use?
- What is a possible, out of the box, approach a student might take?
- How have students in the past approached this task?
- What are some typical conceptions students bring to this task that could lead students astray?
- What are students' prior knowledge that this task requires?

Because this thinking occurs during the planning phase rather than during the live action of a classroom, in theory, teachers have time to consider these types of questions. That's easier stated than what is often the reality for teachers.

That said, spending time on your own, or better yet with a colleague during a PLC, to anticipate student thinking on a task is the foundation for orchestrating meaningful mathematical discourse (one of the mathematics teaching practices from Principles to Actions, NCTM 2014).

To see how I began anticipating student thinking on the "Ticket Booth" problem, go here. I welcome your thoughts.

]]>What does it mean to "Do Math"? When students are *doing* math, what does it look like? Does it look different at different grades? For different math topics? When you're in school compared to outside of school?

The Common Core Standards for Math helps provide a vision of what it means to "Do Math." Let's look closely at the Common Core Standards for Math.

**The Common Core Standards for Math are made up of two sets of standards.**

*Standards of Math Content*: The K-12 math concepts and skills broken into domains of mathematics (e.g., number and operation, algebra, funcitions, statistics and probability, etc.) depending on the grade level. These standards progress from grade-to-grade to build a cohesive network of topics within a structure of mathematics.*Standards of Math Practice*: The Math Practices are the "habits of mind" of doing mathematics. The Math Practices are the*actions*(i.e., behaviors, thinking, etc.) people do with the*things*we call math content.

**When we say "Do the Math", I think of the Math Practices**.

The eight Standards of Math Practices are the same for all grade levels K-12, which become more fully developed as the doer of math matures.

As teachers implement CCSS math, focusing on the Standards of Math Practice is vital to our transitioning from the old standards, assessments and instructional practices to those that students need to be college and career ready. **I'm not the only one seeing the importance of shifting our instruction to ensure that students are engaging in the Math Practices.**

**The Math Practices and Smarter Balanced**

Smarter Balanced assesses students' performances of Standards of Math Practices. Check out the Claims for the Mathematics Summative Assessment, which describes the four claims that Smarter Balanced will find evidence for as students complete their grade level assessments. **Notice the direct connection to the Standards of Math Practices.**

**The Math Practices and Higher Education**

Institutes of higher education highlight the importance of the Standards of Math Practices. Check out the University of California Board of Admissions and Relations with Schools (BOARS) Statement on High School Mathematics Curriculum Development under the Common Core State Standards. BOARS is the organization of that oversees the UC A-G course approval process.

**There are two sentences in the statement that really struck me:**

- In the first paragraph, there is this one...."Developing a coherent mathematics curriculum that is fully consistent with the CCSSM will involve much more than simply reordering topics to be covered."
- In the last paragraph, there is this one...."It is BOARS’ expectation that courses developed in accordance with either sequence will receive subject area “c” approval provided that they satisfy the course requirements for area “c” presented in the A-G Guide and that they support students in achieving the Standards of Mathematical Practice given in the CCSSM."

These statements tell me that to receive UC course approval for new math courses, we will need to demonstrate that courses and instruction will **engage students in the Standards of Math Practices on a real and frequent basis**.

As we continue to transition to Common Core Math, we need to keep the Standards of Math Practice at the forefront of our professional work, including our professional learning. **We should strive to know deeply the Standards of Math Practice. **

- What are they?
- What do they look like in the classroom?
- How do we continue to shift our teaching practices to engage students in them (see Principles to Actions)?
- How do we support our district and site leadership to understand the shifts as well (see Principles to Actions)?

**We want our students to be active doers of math, not innocent bystanders of math.**

What are your thoughts on this? Please share.

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*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.*

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?

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.

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