Focus on the Standards for Mathematical Practice

Focus on the Standards for Mathematical Practice
Focus : This professional development focuses on the Standards for Mathematical Practices as they relate to students and how they connect to the NCTM Principles to Actions Effective Teaching Practices. Participants will dive into the Standards for Mathematical Practice to deepen their own understanding, discuss teaching practices to elicit them in students, modify tasks to highlight the practices and reflect on their own beliefs to identify where they are in the continuum from teacher-directed to student-centered instruction. Materials Needed: Print maze handout: participant devices (if using Kahoot for pre-assessment), chart paper, markers, note cards, sticky notes, cards ace-8 from a deck of cards (remove 9s, 10s, and face cards) Links for other handouts are embedded either on slides or in leaders’ notes. Before the session, post the group's Community Norms poster if this is part of your professional development routine. Remind participants that they can add and revisit this poster as needed.

Source: www.amazeaweek.net
Welcome! Complete the Mazes on Your Table as You Wait for the Session To Begin Plan about 10 minutes for slides Link for maze: As teachers arrive, ask them to complete the mazes on their tables. One maze has one starting point, one ending point, and one correct path. The other has multiple entry/exit points and therefore multiple solutions. We will revisit these in this professional development. Source:

Learning Objectives The overarching goals for this professional development are to: Deepen our understanding of the K-12 Standards for Mathematical Practice. Make connections between the Standards for Mathematical Practice and how we can help develop these standards and support mathematics learning for all students. The content standards define what students should understand and be able to do at each grade level, whereas the SMP ensure that the “processes and proficiencies of mathematics” are at the heart of teaching and learning. These two types of standards are interconnected; interacting to improve instruction and learning for all students. This professional development uses the lens of the SMP to guide teachers through the processes of teaching and learning of mathematics. Throughout this session, teachers develop a deeper understanding of the eight mathematical practices and reflect on why they are important for students.

Tell Us What You Know! Get your devices and
let’s take a pre-assessment! This is a quick pre-assessment to gauge where participants are with their understanding of the Standards for Mathematical Practice. There is a Kahoot for this. For those who do not wish to use Kahoot, the questions are provided on the next 4 slides. Skip these slides if you use Kahoot. Kahoot directions: Facilitator goes to getkohoot.com Username: Tools4Teachers Password: Tools4Teachers Click on My Kahoots (in black at the top of the screen). Click on “Shared with Me” Choose Standards for Mathematical Practice Kahoot Click Play This will bring up a new screen that you want to project. Choose Classic: “Player vs. Player” 1:1 Devices Under Game Options, all choices should be off. If the first four (these deal with points and randomized order of questions answer) are “ON”, click the box to turn them “OFF” A Game PIN will appear on the screen. Instruct participants to follow the directions on the screen. (go to kahoot.it and enter the game PIN in on their devices to join the Kahoot), Each person chooses a nickname (can be anything - does not have to be their real name). After all participants have joined (there is a count on the left hand side of the screen), facilitator clicks “Start” and the Kahoot begins. Each question will show on the screen, and participants select their choice on their device. The question will display on the projection screen to give participants a chance to read it. Then the question and the answer choices appear on the screen. The question nor answer choices appear on the participants’ devices - only the color/shape that correspond with answer choices on the screen. The first question is a practice question to familiarize teachers with Kahoot. After the answers are displayed, the group can discuss the question. The facilitator clicks next to proceed to the next question when he/she is ready.

Pre-Assessment Rate your understanding of the Standards for Mathematical Practice: I could teach this session. I can identify and give examples of them. I can tell you what they are. Huh? What are the Standards for Mathematical Practice? The next 4 slides have the questions from the Kahoot preassessment. This is a question from the Kahoot preassessment. Skip /delete this slide if you do the Kahoot.

Pre-Assessment The Standards for Mathematical Practice are for:
AIG students High school students All students Both a & b This is a question from the Kahoot pre assessment. Skip /delete this slide if you do the Kahoot.

Pre-Assessment The Standards for Mathematical Practice define the content students learn at each particular grade level. True False This is a question from the Kahoot pre-assessment. Skip or delete this slide if you do the Kahoot.

Pre-Assessment The Standards for Mathematical Practice describe what it means to know and use mathematics. True False This is a question from the Kahoot pre-assessment. Skip / delete this slide if you do the Kahoot.

Let’s Explore the Mathematical Practices
In this activity, you will work with others to deepen your understanding of the assigned Standards for Mathematical Practice through exploration of the resources provided and present your learning to the group. What do you notice about the first three words in the description of each Standard for Mathematical Practice? Plan about 20 minutes for slides Download descriptors from create a handout. Participants will use this handout to build on their previous understanding of the Practice Standards. Take a look at the first resource together (the SMP from CCSS). What do you notice about the first three words in the each of the descriptions for the Standard for Mathematical Practice? They are the same three words: “Mathematically proficient students…” The descriptions continue to explain what students say and do as they demonstrate understanding of that practice. However, in the document, some of the examples are middle and high school and not exactly realistic for elementary classrooms. The NCDPI documents help us with that. Facilitator assigns each pair (group) of participants one of the SMPs (8 groups).

Another Look at the Mathematical Practices
On the chart paper, generate classroom practices for your grade level that exemplify this standard. What do teachers do: BEFORE a lesson to provide opportunities for this SMP to happen? DURING a lesson to cultivate the SMP? What do the students do? Example - What does this practice look like in action? Specifically, ask participants to note and create a chart to share that has the following information about their assigned Standard for Mathematical Practice: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. Give teachers about minutes to work on this, then give the next instructions to modify the problem (next slide).

Consider This Task Think about how you might revise this task to focus the Standard for Mathematical Practices as they are presented. Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? After participants are finished with their posters, before they share what they learned, ask them to consider the task on this slide. What do they like about it? What would they change? After each group shares their practice, we will put our deeper understanding to practice as we consider how to modify the following problem to emphasize each Standard for Mathematical Practice. This sets the purpose for participants not presenting their task to listen through the lens of how they could implement the SMP in their classroom through this task. If participants struggle, there is a modification suggestion on a slides behind each SMP that you could use to guide them or use as another example.

Sharing What We Learned
Each group reports on the discussion in their group regarding their assigned practice. This activity should take about 60 minutes (7-8 minutes per practice). What the teacher does to develop these practices/habits of mind in students (what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. Encourage those not presenting to take notes on the new learning or key ideas of the Standards for Math Practice being shared. Depending on the participants’ overall understanding of the Standards for Mathematical Practice, facilitators may vary the methods they ask participants to use to report their discussions and what they learned. They may also vary the amount of time they spend on this task (more if teachers do not have a working understanding of the practices, less if participants have had lots of previous experience with putting the practices into action). Facilitators may choose for each group to share what they learned and their answers to the questions posed with the large group or they may choose to have teachers write the information on chart paper for a gallery walk, with specific instructions about how to interact with the charts (place a sticky note with a ! on it for something that strikes you as interesting, make a connection to your classroom practice and write it on a sticky note and place on the chart, write a question you have for the group on a sticky note and put it on the chart, etc.) .

SMP 1: Make Sense of Problems and Persevere in Solving Them
Students should be able to: Explain the meaning of the problem. Students may use concrete objects and/or pictorial representations. Come up with a strategy for solving the problem. Identify the connections between two different approaches to a problem. Determine whether or not the solution makes sense. The following slides are available to provide more info and can be displayed as groups present their work (to ensure they hit major points). If teachers are very knowledgeable of the Standards for Mathematical Practice, the slides can reviewed quickly or may be skipped. This slide is to display as group 1 presents their poster (to ensure they hit major points of this Math Practice Standard). The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action?. For more ideas: Connection to Effective Teaching Practices: If we want students to make sense of problems and persevere in solving them, then teachers must implement tasks that promote reasoning and problem solving and support productive struggle in mathematics.

What Problems? What types of problems do elementary students make sense of? This chart, referenced in the NC Math Standards, represents common addition and subtraction situations. The chart can be explored in detail in later professional development.

(“How many in each group?” Division) Number of Groups Unknown
Unknown Product Group Size Unknown (“How many in each group?” Division) Number of Groups Unknown (“How many groups?” Division) 3 × 6 = ? 3 × ? = 18, and 18 ÷ 3 = ? ? × 6 = 18, and 18 ÷ 6 = ? Equal Groups There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? Arrays,2 Area3 There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example. What is the area of a 3 cm by 6 cm rectangle? If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it? If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it? Compare A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long? A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? General a × b = ? a × ? = p, and p ÷ a = ? ? × b = p, and p ÷ b = ? This chart, referenced in the NC Math Standards, represents common multiplication and division situations. The chart can be explored in detail in later professional development.

Persevere? Think about a “maze moment” you’ve experienced in math (as a student or as a teacher). Depending on the audience and morale, you way want to encourage participants to stand up and find a partner to discuss the next questions with or you may want to have them write a short reflection about their maze moment, or you may just as them to think about it and ask for volunteers to share. How did you solve it? How did you know where to start? ...where to end? What strategies do you use? What do you do when you come to a dead end? How do you decide which path to follow? Usually, we try something, figure out if it works, reevaluate, try something new, and repeat until successful. That’s problem solving...whether it’s a maze or a math task, in a classroom or in a relationship. The power of the “maze moment” language is that it explicitly teaches students to anticipate being stuck from time to time while solving problems. It reinforces a growth mindset and emphasizes that mistakes are a valuable part of the learning (and growing) process. Think about a “maze moment” you’ve experienced in math (as a teacher or as a student).

How are the mazes similar? How are they different?
Reflect on the mazes. How are the mazes similar? How are they different? How are your strategies for solving them similar? How are your strategies different? How do your strategies change if our maze has multiple entry points?...and multiple exits? Show maze slide… Ask teachers to reflect on the mazes they did at the start of this module. How are the mazes similar? How are they different? How are your strategies for solving them similar? Different? How do your strategies change if our maze has multiple entry points?...and multiple exits? Source

How does this relate to the tasks we give students?
Where did you start? Where did you end? Which way is right? How does this relate to the tasks we give students? Consider this maze… Where did you start? Where did you end? Which way is right? Does it matter? How does this relate to the tasks we give students? Source

Enhance a Task: SMP 1 Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? How can we enhance this task to focus on SMP 1? Ask all participants to suggest modifications that would highlight SMP 1. Click to next slide for one possible modifications to highlight this practice.

Enhance a Task SMP 1: How can we describe the story problem? Can we act out the situation? Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? One possible way to modify implementation of this task is to ask students to describe the story problem or act out the situation.

Additional Ideas for Implementing SMP 1
Question the Question - analyze questions and tasks for complexity; potential for rich discussion and multiple strategies. Open-Ended - ask open-ended questions before, during, and after the task to promote problem solving. Anchor Charts - create anchor charts of students’ solutions; can feature multiple entry points to the problem, sequence discussion, prompt students who are “stuck,” and challenge students to show the solution multiple ways. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 1 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in An NCTM resource provides ideas:

SMP2: Reason Abstractly and Quantitatively
Students should be able to: De-contextualize – comprehend a given situation and represent it symbolically. Contextualize – consider the referents for the symbols they are working with. Understand the meaning of the quantities, not just how to compute them. This slide signals that it is time for group 2 to present their work. Display it in the background as they share their poster and to ensure they hit major points of this Math Practice Standard. You do not need to read the slide if they hit the points. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connection to Effective Teaching Practices: If we want students to reason abstractly and quantitatively, then teachers must pose purposeful questions.

Enhance a Task: SMP 2 Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? How can we enhance this task to focus on SMP 2? Ask participants to suggest modifications that would highlight SMP 2. Click to next slide for one possible modifications to highlight this practice.

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? SMP 2: Decontextualizing: Write an equation that represents the story and solve Contextualizing: Label the units in your solution. Make explicit the equation used to model the story may be different from the equations used to solve. The goal is to make connections between the story and the equation. We decontextualize when we put the story situation into an abstract form (represented by an equation). That involves replacing context with numbers and symbols. After solving it, re-contextualizing the mathematics includes referring back to the story to understand the quantities context.

Number Webs - Students given a quantity and asked to use many different ways to express that quantity. Headline Stories - Students write a story problem for a given expression. Pinch Cards - Students indicate (through cards, interactive technology or signals) the operation that matches a problem. Notice/Wonder - Have students discuss what they notice about this problem and what they wonder. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 2 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

SMP 3: Construct Viable Arguments and Critique the Reasoning of Others
Students should be able to: Make conjectures. Use counterexamples in their arguments. Justify their conclusions and explain them to others. Listen and/or read other’s arguments and determine if they make sense. Ask questions to get clarification of an explanation. This slide is to display as group 3 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connection to Effective Teaching Practices: As students represent, explain, critique and justify their thinking and the reasoning of others, they engage in reasoning and sense-making. Principles to Action: Ensuring Mathematical Success for All calls for teaching practices that “facilitate discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments (NCTM 2014, p. 29) This practice can also involves explaining reasoning and constructing argument in writing. Often times math journals are used for this. In the process of constructing arguments, students can identify their own misconceptions and/or errors. As other students share their strategies, listeners build their repertoire of strategies and mathematical ideas by evaluating the reasonableness and by connecting it to their thinking. If we want students to construct viable arguments and critique the reasoning of others, then teachers must pose purposeful questions and facilitate meaningful mathematical discourse.

Why Math Talk? As we represent, explain, justify, agree and disagree, this shapes the way we learn mathematics. It: Engages students. Reveals understandings (and misunderstandings). Supports robust learning by boosting memory. Supports deeper reasoning. Supports language development. Supports development of social skills. Number talks are deeply connected with SMP 3. What if students don’t talk? Asking probing questions promotes student discourse. How did you get that answer? Why did you choose this method? How did you know which operation to use? Why does that work? Will it always work? Can you draw a picture to prove it? How are these problems different? ...the same?

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? How can we enhance this task to focus on SMP 3? Ask participants to suggest modifications that would highlight SMP 3. Click to next slide for one possible modifications to highlight this practice.

Enhance a Task SMP 3: Add “Explain how you know” to the task.
Add think - pair - share to the lesson. Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? Explain how you know. If your goal was to highlight SMP 3, what change could you make? We could add “explain how you know” to the end of the task or add a think-pair-share to the lesson. Are there other ways? After students have explained their thinking, you could ask students to choose someone’s written strategy and agree or disagree. A teacher could “plant” a faulty reasoning solution and ask students to identify and correct mistakes.

Eliminate It - Students are presented with 4 mathematical concepts; decide which one should be eliminated and tell why. Agree or Disagree - Students are taught to respectfully agree or disagree with each other’s thinking and explain why. My 2 Cents - Students given sample arguments with flawed reasoning. They work in groups to identify error and improve the argument. Number Talks This slide is included to help give other classroom practices to bring SMP 2 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann.

Understanding the Standard
What do we do in the classroom to get students to justify their answer and defend their process for finding the answer? How do we help students understand math skills and concepts so they can construct viable arguments? How do we help students consider and judge the reasonableness of other answers and strategies? Participants reflect on these questions and discuss at their tables before sharing with the whole group.

Listening and Speaking to Understand
Use what she just said to adjust your answer. Who can tell me what ___ just said? Who can repeat what ___ just said? Who can share what they heard someone else say? Turn and Talk to a neighbor…as you talk be sure to _______ as you share listen for ___________. This is what it might sound like to talk or listen for understanding…watch/listen. Who can model this for the class? These prompts can be used to encourage listening and speaking to understand. What other prompts would you add to this list?

Assertion vs. Argument Assertion: a statement of what students want us to believe without support or reasoning. The answer is correct “because it is,” “because I know it,” or “because I followed the steps.” Argument: a statement that is backed up with facts, data, or mathematical reasons. Constructing viable arguments is not possible for students who lack an understanding of math skills and concepts. SMP states that “mathematically proficient students construct viable arguments.” What is a mathematical argument? How is that different from an assertion? Emphasize the last bullet point. How is constructing a viable argument different that just talking about mathematics?

SMP 4: Model with Mathematics
Students should be able to: Apply the mathematics they know to solve everyday problems. Use equations, graphs, tables, diagrams, etc., to show the mathematical relationships in their model. Think about whether the model they have created makes sense and modify if necessary. This slide is to display as group 4 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connection to Effective Teaching Practices: If we want students to model with mathematics, then teachers much use and connect mathematical representations.

What is NOT Modeling with Mathematics?
“As elementary teachers, we have misinterpreted the term model to simply mean the use of manipulatives, a misunderstanding that is causing our students to miss the mark when it comes to modeling with mathematics. The questions then arise: What is modeling with mathematics? How can we make a cognizant effort to be sure it is taking place in our classrooms?” Teaching Children Mathematics, Graham Fletcher - April 11, 2016 Marzano says that in order to fully understand something, we also need to understand what it is not. Graham Fletcher, in his April 11, 2016 post on the NCTM blog: In short, the use of manipulatives does not ensure that modeling with mathematics is taking place. If the mathematics is not contextualized, modeling with mathematics cannot exist.

What is NOT Modeling with Mathematics?
does not mean “I do, we do, you do.” Teaching Children Mathematics, Graham Fletcher - April 11, 2016 The gradual release body of research is specific to reading and should not be generalized to mathematics instruction. In mathematics, the research points us to flipping that model (you do - we discuss and learn - I (teacher) provide support, we discuss and make connections. Traditional word problems that give all the information needed lend to students locating quantities in the problem, identifying key words, applying operations and moving on to the next problem. Modeling activities ask both the teacher and students to approach mathematics differently.

What IS Modeling with Mathematics?
If mathematical modeling is used to interpret real-world situations in mathematical formats, is writing an equation to represent a problem enough? What if we ask students to: Identify the problem (or pose a question)? Make an estimate? Identify the variables needed to solve and answer the problem or question posed? Fletcher suggests that modeling requires more than just a simple equation. He suggests three actions that ensure our students are engaged in modeling (on slide). Contrast with “I do, we do, you do.” Modeling does take different forms according to the grade level and complex mathematics.

Mathematical Modeling
Mathematical modeling emphasizes the fact that “thinking mathematically” is about interpreting situations mathematically at least as much as it is about computing. (Lesh and Lehrer 2003, p. 111) Lesh, Richard, and Richard Lehrer “Models and Modeling Perspectives on the Development of Students and Teachers.” Mathematical Thinking and Learning 5 (2 and 3): 109–29.

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? SMP 4: How can you model this situation? Compare your model with a partner’s model. How are they similar? How are they different? Students can model this story problem in different ways (drawing, manipulatives, equations, with words, etc.) As students compare their model to a partner’s, they make connections between different mathematical representations, which also highlights the Effective Teaching Practice from NCTM’s Principles to Action. We could ask students to write an expression to represent the number of cards in the situation if your instructional focus was on modeling mathematics with expressions. BUT, remember we discussed earlier that perhaps writing an expression to model the problem is insufficient, so we will explore another modification that models it differently (with a table) after practice 7.

Anchor Charts - Create charts of various representations and models that students use to solve a problem. Bar Diagrams - Help students visualize problems and make sense of them. Technology Tools - Electronic manipulatives such as Illuminations allow for exploration and maneuvering tools. Question the Model - Encourage students to analyze models for effectiveness and efficiency. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 4 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

SMP 5: Use Appropriate Tools Strategically
Students can: Consider which available tools they might use when solving a problem. Recognize the strengths and limitations of the tools they are using. Identify additional external resources, such as a website. This slide is to display as group 5 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connection to Effective Teaching Practices: If we want students to use appropriate tools strategically, then teachers must use and connect mathematical representations.

Which tool(s) would you use to help you solve this problem?
Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? SMP 5: Which tool(s) would you use to help you solve this problem? This practice is heavily connected to modeling mathematics. There are many tools students to use to solve this problem, and those may vary depending on the student’s level of mathematical reasoning and the teacher’s goal for the task. Students at a concrete level may need physical models (snap cubes, base ten materials, square tiles) to solve, some may prefer to use open number lines or graph paper but still other students may use a calculator to find the solution. Free access to tools is recommended so that students can chose the model that is meaningful for them.

Broken Ruler - Give students rulers that are “broken” at either end (either don’t begin with 0 or end with 12) to measure objects. Exploring a Magnified Inch - Students fold a sentence strip to mark ½, ¼, and ⅛ then explore and discuss concepts related to fractions on a number line, equivalent fractions, and measurement techniques. In My Head - Given a story problem, students decide if solution should be found mentally, on paper, or with a calculator. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 5 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

SMP 6: Attend to precision
Students should be able to: Communicate precisely to others. Use clear definitions in discussion. Explain to each other. Explain the meaning of the symbols they choose. Specify units of measure. Calculate accurately and efficiently. This slide is to display as group 6 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connections to Effective Teaching Practices: If we want students to attend to precision, then teachers must facilitate meaningful mathematical discourse, use precise language and “mathematizing” informal language, and elicit and use evidence of student thinking to help students build precise language.

Precision ≠ Perfection
We help students build precision as we uncover meaning and connect their ideas and claims (informal language) to formal math language. Look for, acknowledge and celebrate precise claims, even if the language is not perfect. Precise claims can be expressed in imperfect language. Some brilliant mathematical ideas aren’t initially communicated with very precise language. There’s not always a perfect word. Christine Newell’s NCTM 2017 Ignite Talk Ask teachers to consider these points from Christine Newell’s NCTM 2017 Ignite Talk. We aren’t going to watch the video in this session, but if you’d like to explore further, here’s the link: Expecting perfect language can impede students’ emerging math ideas. We have students who say nothing in math class because they are afraid they don’t have the perfect words. What are the implications for teachers? We have to work to uncover the mathematics, no matter what language students bring. What does it mean to “Let language be student led?”

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? SMP 6: Does this problem require an exact answer or is an estimate enough? Explain your reason. Sometimes we need very precise answers; other times an approximation will do. Asking students to determine if an estimate is sufficient requires them to consider the intent of the problem or the situation. Another modification for this task would be to ask students how they would know if their answer made sense. They could use estimates or benchmarks to approximate the answer before computing and compare the estimate to their solution to assess the reasonableness.

Word Webs - Students make web of related words to a math term. Word Walls - Highlight key vocabulary in a prominent location in the classroom. Various activities can be used throughout the year to strengthen understanding. Sort and Label - Identify similarities and differences between terms. Math Journals - Students reflect and write about mathematical understanding using precise language. Translate the symbol - Students represent math ideas with numbers and symbols (e.g. subtraction, add, multiplication, combine). O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 6 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

SMP 7: Look for and Make Use of Structure
Students should be able to: Look closely to identify a pattern or structure. Step back for an overview and shift perspective. See complicated things as single objects or as being composed of several objects. This slide is to display as group 7 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connections to Effective Teaching Practices: If we want students to look for and make use of structure, then teachers must establish math goals to focus learning, use and connect mathematical representations, and build procedural fluency from conceptual understanding.

Enhance a Task: SMP 7 Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? How can we enhance this task to focus on SMP7? Ask participants to suggest modifications that would highlight SMP 7. Click to next slide for one possible modifications to highlight this practice.

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? What if Jack started with 6 (9, 12) boxes of cards (with 10 in each box)? SMP 7: Vary the values to explore the structure of the problem. Ask, “What is changing? What is staying the same?” Another suggestion is to vary the values to explore the structure of the problem - to change the number of boxes Jack has to focus on SMP7 (look for and make use of structure). As students explore the structure of the problem by varying the number of boxes he starts with, there is an opportunity for discussion about what is changing and what is staying the same (and why).

Intentional exposure to patterns - Feature numerical, geometric, and growing patterns in class. Use of Charts (Hundreds, Multiplication, Addition) - Highlight patterns of even/odd, multiples of benchmark numbers, directional meanings, etc. Use of Ratio Tables - Use table to help organize data and look for patterns and functions. Property Exploration - Provide tasks where students explore properties of addition and multiplication. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 7 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

SMP 8: Look For and Express Regularity in Repeated Reasoning
Students should be able to: Notice if calculations are repeated, and look for both general methods and for short cuts. Evaluate the reasonableness of their results. This slide is to display as group 7 presents their poster (to ensure they hit major points of this Math Practice Standard). It can be skipped. The facilitator ensures that the group presents on all three questions asked of the group: What the teacher does to develop these practices/habits of mind in students (before and during a lesson - what teacher moves does he/she use and/or what questions does he/she ask), What the students say and do, and What lessons, ideas or task examples bring this practice to life in their grade level. What does this practice look like in action? Reference for more classroom examples. Connections to Effective Teaching Practices: If we want students to look for and express regularity in repeated reasoning, then teachers must establish math goals to focus learning, implement tasks that promote reasoning and problem solving, use and connect representations, and build procedural fluency from conceptual understanding.

Enhance a Task: SMP 8 Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 5 baseball cards. How many cards does Jack have now? How can we enhance this task to focus on SMP8? Ask participants to suggest modifications that would highlight SMP 8.

Give various values and talk about patterns.
Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack some baseball cards. How many cards does Jack have now? What if: Sean gives him 6 cards? Sean gives him 8 cards? SMP 8: Replace 5 with ‘some’ Give various values and talk about patterns. If the focus is on SMP 8 (Look for and express regularity in repeated reasoning), you could ask students what happens if we replace the number of baseball cards Sean gives Jack with the word “some.” This opens up the opportunity for a discussion (also SMP3) about the patterns noted.

Enhance a Task Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? What if Jack started with 6 (9, 12, n) boxes of cards (with 10 in each box)? Additionally, to extend the exploration of the structure and to integrate modeling to a higher level that just writing an equation, you could also incorporate some mathematical modeling and ask students to create a table to model the change and extend it to generalize for n boxes of cards. Teachers will actually engage in this task. Show the text on the left side. Ask teachers to solve with the modification suggested in green. What SMPs are involved? Think about how you solved it...did you make a table (model with mathematics SMP4), look for and analyze structure (SMP7), and/or look for regularity in repeated reasoning? Click to next slide to reveal the mods we could make and the SMP connections. Give participants about 2-4 minutes to solve and discuss.

Enhance a Task SMP 4, 7 & 8: Use a table to model
Task: Jack has 3 boxes of baseball cards. He has 10 cards in each box. Sean gives Jack 12 baseball cards. How many cards does Jack have now? What if Jack started with 6 (9, 12, n) boxes of cards (with 10 in each box)? SMP 4, 7 & 8: Use a table to model Can we extend this thinking to generalize for n boxes of cards? Think about how you solved it...did you make a table (model with mathematics SMP4), look for and analyze structure (SMP7), and/or look for regularity in repeated reasoning? Click to next slide to reveal the mods we could make and the SMP connections.

Organize and Displaying Data - Use graphs, tables, and other graphics to help students look for patterns and discover rules. Careful Task Selection - Choose tasks that orchestrate mathematical investigations so that students are led to discover shortcuts, rules, or generalizations. Broken Calculator - Create tasks where students have to create even or odd numbers without using them on the calculator. O’Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action, Heinemann. This slide is included to help give other classroom practices to bring SMP 8 to life. It may or may not be needed, depending on how thoroughly the participants address this component of the task. If not needed, it can certainly be skipped. These additional ideas for implementing the Standards for Mathematical Practice are all included in

Quiz - Quiz - Trade Select a card. The number on the card is your Mathematical Practice. Write quotes you have heard (or that you could hear) from students in your classroom that exemplify this practice. Quiz - Quiz - Trade Adapted from Jenny Bay Williams’ presentation at the K-8 Meredith Mathematics Leadership Institute For this activity, you need cards This can be a deck of cards from which you’ve removed all 9, 10, and face cards except Aces, which would be 1. You could also have slips of paper numbered 1-8 that participants can choose from. Participants write quotes that they’ve heard from students that exemplify their practice, then participants find a partner. Partner A reads his/her card (not disclosing which SMP it is). Partner B guesses the SMP. Repeat - Partner B reads his/her card. Partners trade cards then find a new partner.

The Practices in Action
A slab of soap on one pan of a scale balances ¾ of a slab of soap and a ¾ pound weight on the other pan. How much does the full slab of soap weigh? Next, participants will watch a video of the SMP in action as students solve this problem. Give participants 2 minutes to consider the problem, how they would solve, and how they expect students to solve it.

Let’s Take A Look! Play the 5 min video, which illustrates a SMP. (1,2,4,6) This is a fourth grade task that highlights SMP 6 and using concrete models to solve a problem using fractions. As participants watch the video, have them use the NCTM “Look for” document and their posters to identify SMPs in the video.

If we want students to... Then teachers must...
Just as there is a specific structure to the content standards (domains, clusters, and standards), there is a structure to the Standards for Mathematical Practice (graphic on left). This graphic comes from the work of William McCallum to explain the structure of the 8 Mathematical Practices. These 8 practices are defined on pages 5-7 of the Common Core State Standards. Dr. McCallum noted, “if you try to do everything all the time, you end up doing nothing.” Rather than saying we are doing every practice every day, it makes sense to find places in our instruction where we can focus on developing a specific practice. Connecting SMP to the Principles to Action teaching practices allows us to develop the SMPs in students. Let’s explore this a little deeper. These practices are the foundation for a paradigm shift in mathematics. If our goal is to help students develop these skills (who can think mathematically, use tools to make sense of problems, persevere the answer isn’t obvious at first glance, and communicate their thinking with precision), the Mathematical Teaching Practices we explored earlier in this PD are the vehicle to elicit these behaviors.

Continuum of Effective Teaching Practices
Teacher provides students with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Teacher presents math in small chunks, with help provided, so that students arrive at answers quickly and without high-level thinking. How can we provide optimal learning opportunities for students to become “mathematically proficient?” Consider (Discuss) this continuum. How does the teacher’s role change as we move across the continuum? How does the student’s role change? As we move from left to right, the focus is less on teacher-centered instruction and more student-centered. Duplicate the continuum on chart paper for the next activity.

Teacher provides students with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Teacher presents math in small chunks, with help provided, so that students arrive at answers quickly and without high-level thinking. Have these teacher moves written on post-it notes. Participants place the sticky notes on the continuum in the room and discuss why they placed each where they did.

Teacher provides students with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Teacher presents math in small chunks, with help provided, so that students arrive at answers quickly and without high-level thinking. Teacher provides open number line for students to use. “What patterns do you notice across these problems?” “Remember PEMDAS and do steps in that order.” “Wow! You solved that so quickly.” “You changed your strategy. What made you want to try something different?” For each teacher move, indicate what SMP students might have the opportunity to demonstrate. What do you notice about the connections between the Principles to Action Effective Teaching Practices and the Standards for Mathematical Practice? Each group writes a teacher move (or Effective Teaching Practice from earlier modules) that corresponds with their assigned SMP and posts it on the continuum in the classroom.

Time to Reflect Choose 2 Standards for Mathematical Practice. How do you plan to connect the mathematics content with these Standards for Mathematical Practice in your classroom? What from this module resonated with you? Provide some time for reflection and for participants to capture what they want to remember and put into practice from this module.

References A Maze A Week. (2014, October 7). Retrieved May 23, 2017, from Bay-Williams, J. (2016, July 27). Leading for Mathematical Proficiency: Making Explicit Connections to Student Outcomes. Lecture presented at K-8 Meredith Mathematics Leadership Institute at Meredith College, Raleigh, NC. Illustrative Math. (2014, August 1). CCSS - Soap {video file}. Retrieved May 23, 2017, from National Council of Teachers of Mathematics. (2014). Principles to actions: ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for mathematics: Kindergarten introduction. Retrieved from O'Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action: Implementing the Common Core Standards for Mathematical Practice, K-8.

Grade Level Writing Teams
Kindergarten: Dawne Coker Carol Midgett Leigh Belford Lynne Allen First Grade: Felisia Gulledge Martha Butler Laura Baker Danielle Long Second Grade: Tery Gunter Carly Morton Diane Wells Isaac Wells Third Grade: Leanne Daughtry Robin Hiatt Meg McKee Kaneka Turner Fourth Grade: Ana Floyd Lisa Garrison Shelly Harris Deanna Wiles Fifth Grade: Marta Garcia Brandi Newell Susan Copeland Rebekah Lonon Lead writers are green….

Tools for Teachers Staff
Kelly DeLong, Co-PI and Project Manager Kayonna Pitchford, Co-PI and IHE Janet Johnson, Outside Evaluator Jeane Joyner, IHE and Reviewer Katie Mawhinney, IHE and Reviewer Drew Polly, IHE and Reviewer Wendy Rich, K-5 Coordination and Reviewer Catherine Stein, IHE and Project Liaison Please give appropriate credit to the Tools for Teachers project when using these materials.

Focus on the Standards for Mathematical Practice

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