Number Talks: Strengthen Students’ Mathematical Reasoning Robin Moore K-6 Math Specialist James Morris School, RSD#6 Sara Baranauskas K-6 Math Coordinator Regional School District #6

About Us…. Robin Moore Sara Baranauskas K-6 Math Specialist, James Morris School 12 years Grades 1 and 5, 2 years Math Specialist Email: rmoore@rsd6.org twitter: @rmoore628 Sara Baranauskas K-6 Math Coordinator, Regional School District No. 6 15 years Grades 5 and 6, 3 years Math Coordinator Email: sbaranauskas@rsd6.org twitter: @sarabara13

Today’s Purpose To provide teachers with an instructional strategy that is grounded in developing students’ mathematical reasoning and conceptual understandings rather than just teaching rules and procedures without understanding of numerical relationships.

What is Mathematical Reasoning? As one of the five Process Standards (NCTM 2000), reasoning (and proof ) is an essential part of acquiring and using mathematics. More than a means by which to confirm whether a solution is correct, mathematical reasoning involves exploring the mathematics at hand; generating, implementing, and evaluating conjectures; as well as justifying our thinking and actions as we engage in mathematics. Simply put, mathematical reasoning is about making sense of the mathematics in our world as well as making mathematical sense of our world. Nurturing Mathematical Reasoning, Jennifer Thom, Teaching Children Mathematics, Nov 2011

Focus Questions How can Number Talks develop students’ mathematical reasoning? How do Number Talks align to the Common Core Mathematical Practices and Content Standards?

What is a Number Talk? A number talk is a quick (10 min) classroom routine used to develop computational fluency that promotes students to: • Clarify thinking. • Investigate and apply mathematical relationships. • Build a repertoire of efficient computational strategies. • Make decisions about choosing efficient strategies for specific problems. • Consider and test other strategies to see if they are mathematically logical. • Build connections between key conceptual ideas.

Key Components of a Number Talk Classroom environment and community Classroom discussions The teacher’s role (facilitating vs. teaching) Purposeful computation problems The role of mental math

Classroom Environment and Community Setting up takes time and MUST be initiated from day 1. Explicit modeling and reinforcement Set Norms – a common goal for learning and understanding for all Safe, risk free environment. Students know that their trust in you and their classmates is well founded. Increases self confidence Risks often come with learning and learning ALWAYS comes with risks. Predictable routines allow students to feel confident and capable. Effective learning environment will maximize the engagement of learners.

Classroom Discussions Problem is posted for all students to solve mentally. Wait time is critical! Multiple strategies are encouraged. Discrete signals to indicate solution and strategies. Students share, discuss, justify and critique strategies and solutions. Mistakes are an opportunity for learning- wrong answers are used as springboards to address misconceptions.

Classroom Discussions cont’d Benefits of sharing and discussing computation strategies Students have the opportunity to: Clarify their own thinking. Consider and test other strategies to see if they are mathematically logical. Investigation and apply mathematical relationships (connections) Build a toolkit of flexible and efficient strategies. Make decisions about choosing efficient strategies for specific computation problems. Number Talks: Helping Children Build Mental Math and Computation Strategies, Sherry Parrish, 2010

The Teacher’s Role Teacher as facilitator, questioner, listener and learner – NOT teller and explainer. Crafts purposeful problems. Supports discourse to deepen mathematical understanding and knowledge and keeps discussion focused on the important mathematics. Assists students to learn how to structure their comments and wonderings in a meaningful and natural manner. Asks open ended questions – from “What answer did you get” to “How did you get your answer?” and “Why did you choose your strategy?”

Purposeful Problems Teachers choose and develop “just right” mathematical problems and concepts based on student need and objectives. Certain types of problems elicit certain strategies. Random problems do not lend themselves to a common strategy. It doesn’t initiate a common purpose for a Number Talk discussion. Can be differentiated for all students and used for small group settings.

The KEY COMPONENT of any Number Talk The Role of Mental Math The KEY COMPONENT of any Number Talk Focus on number relationships and use these relationships to develop flexible, efficient strategies to develop accuracy (fluency!) Mental math allows students to rely on what they know and understand about number and how they are inter-related. Leads to efficiency with numbers to avoid holding numerous quantities in their heads.

The Role of Mental Math cont’d Strengthens understanding of place value, rather than the discrete quantity of digits. Numbers initially written horizontally – encourages place value understanding. Both formats should be used once students are comfortable with flexible strategies.

Thumbs up indicates solution. Fingers extended signify strategy. Let’s Try It!! 59 + 13 Thumbs up indicates solution. Fingers extended signify strategy.

Our Number Talk What mathematical ideas did we share? How do these strategies encourage mathematical reasoning (number sense and the use of number relationships)?

A Third Grade Number Talk Watch the number talk. Refer back to the Key Components. What do you notice? - Classroom environment and community - Classroom discussions - The teacher’s role (facilitating vs. teaching) - The role of mental math - Purposeful computation problems (Think-Pair-Share)

Let’s apply mathematical reasoning 1. These students share several addition strategies Adding up in chunks Decomposing by place value Making landmark or friendly numbers. How are these strategies similar and different? How could you help students connect among strategies they share? Taken from Number Talks, Parrish

The Content Standards Computational Fluency Number Talks are within the domains of Operations and Algebraic Thinking Numbers and Operations in Base Ten Numbers and Operations – Fractions But Number Talks can also be within any domain of the Common Core content standards.

What about the Practice Standards? MATHEMATICAL PRACTICE 1 Make sense of problems and persevere in solving them Mathematically proficient students explain to themselves the meaning of a problem and look for ways to solve it. apply a solution pathway rather than jumping right into a solution attempt. listen to the strategies of others and apply other approaches. Common Core Standards Initiative (2010)

Teacher Questions to Foster MP 1 How would you describe the problem in your own words? How would you describe what you are trying to find? What do you notice about...? What information is given in the problem? Describe the relationship between the quantities. Describe what you have already tried. What might you change? to this point. What steps in the process are you most confident about? What are some other strategies you might try? How might you use one of the previous problems to help you begin? How else might you organize...represent... show...?

Mathematical Practice 2 Reason abstractly and quantitatively Mathematically proficient students make sense of quantities and their relationships to each other. connect quantity to written symbols. attend to meaning of quantity, not just how to compute them. know and flexibly compute using different properties of operations and strategies. CCSI (2010)

Teacher Questions to Foster MP 2 What do the numbers used in the problem represent? What is the relationship of the quantities? How is _______ related to ________? What is the relationship between ______and ______? What does_______mean to you? (e.g. symbol, quantity, diagram) What properties might we use to find a solution? How did you decide in this task that you needed to use...? Could we have used another operation or property to solve this task? Why or why not?

Mathematical Practice 3 Construct viable arguments and critique the reasoning of others Mathematically proficient students communicate clearly and articulate their ideas to others, their thinking becomes visible and their understanding deepens. follow the thinking of others and communicate why they agree or disagree their understanding often solidifies and strengthens. Math Sense: The Look, Feel and Sound of Effective Instruction, Moynihan (2012) CCSI (2010)

Teacher Questions to Foster MP 3 What mathematical evidence would support your solution? How can we be sure that...? / How could you prove that...? Will it still work if...? What were you considering when...? How did you decide to try that strategy? How did you test whether your approach worked? How did you decide what the problem was asking you to find? (What was unknown?) Did you try another solution that didn’t work? Would it ever work? Why or why not? What is the same and what is different about...? How could you demonstrate a counter-example?

Mathematical Practice 6 More Standards…. Mathematical Practice 6 Attend to precision Mathematically proficient students communicate to each other with precision. attempt to use clear definitions and discussion with others and within their own reasoning –their language is precise, their reasoning can be followed, their words and symbols are clear. Calculate accurately and efficiently. Math Sense: The Look, Feel and Sound of Effective Instruction, Moynihan (2012) CCSI (2010)

Teacher Questions to Foster MP 6 How did you know your solution was reasonable? Explain how you might show that you solution answers the problem. What would be a more/the most efficient strategy? How are you showing the meaning of the quantities? What symbols or mathematical notations are important in this problem? What mathematical language...,definitions..., properties can you use to explain...? How could you test your solution to see if it answers the problem?

Example of Number Talk Progression Kindergarten- Subitizing/Partitioning Dot Images K.CC.B.5, K.OA.A.3

Number Talk Progression Kindergarten- Building to Ten using Ten Frames K.CC.B.5

Number Talk Progression cont’d… Grade 1- Making Friendly Numbers using Double Ten Frames 1.OA.B.4, 1.OA.C.5 1.OA.C.6

Number Talk Progression cont’d… Grade 1- Friendly Number Number Strings 2 + 5 + 8 1.OA.A.2, 1.OA.B.4 1.OA.C.6

Number Talk Progression cont’d… Grade 2- Making Friendly Numbers 9 + 14 13 + 8 + 7 49 + 27 2.OA.B.2, 2.NBT.B.5 2.NBT.B.9

Number Talk Progression cont’d… Grade 3 - Making Friendly Numbers 126 + 38 899 + 25 3.NBT.A.2

Number Talk Progression cont’d… Grade 4 - Making Friendly Numbers 4 x 49 16 x 25 4.NBT.B.5

Number Talk Progression cont’d… Grade 5 - Making Friendly Numbers 840  25 756  24 5.NBT.B.6

Focus Questions How can Number Talks develop students’ mathematical reasoning? How do Number Talks align to the Common Core Mathematical Practices and Content Standards?

Resources The Common Core State Standards Initiative (2010) Moynihan, Christine (2012). Math Sense: The Look, Sound and Feel of Effective Instruction. Stenhouse. http://www.stenhouse.com/shop/pc/viewprd.asp?idProduct=9655 Parrish, Sherry (2010). Number Talks: Helping Children Build Mental Math and Computation Strategies K-5. Math Solutions. http://store.mathsolutions.com/product-info.php?Number-Talks-pid270.html Thom, J (November 2011). Nurturing Mathematical Reasoning: do students physical actions and interactions precede sophisticated reasoning? Teaching Children Mathematics, 234-243.

Free Number Talks Resources Article - Number Talks Build Numerical Reasoning: strengthen accuracy, efficiency and flexiblility…. http://www.mathsolutions.com/documents/NumberTalks_SParrish.pdf Teaching Channel videos https://www.teachingchannel.org/videos/third-grade-mental-math https://www.teachingchannel.org/videos/third-grade-math-solutions Math Perspectives http://www.mathperspectives.com/num_talks.html Insidemathematics.com - Number Talks http://insidemathematics.org/index.php/number-talks Implementing Number Talks Helpful Hints http://www.mathperspectives.com/pdf_docs/num_talks.pdf

Digital Resources Apps Websites Subitize Tree, $.99 Quick Images, $.99 Counting Together, $1.99 Websites Math Coach’s Corner http://mathcoachscorner.blogspot.com/ Stanford Online - “How Children Learn Mathematics”, Spring 2014http://online.stanford.edu/

Free Formative Assessment Tools Show Me – records voice and actions; saves to website for easy linking to websites, digital portfolios, etc Educreations & Screen Chomp interactive whiteboard apps that records voice and actions Three-Ring– take pictures, videos of students, write notes, add to students’ folders.