Silicon Valley Mathematics Initiative 6th – 8th Grade Breakout Sessions Brentwood • Byron • Liberty Coaching Institute October 7th – October 9th and Collaborative Tasks April Cherrington Silicon Valley Mathematics Initiative www.svmimac.org

Agenda Welcome Agenda/Norms Number Talks MARS Tasks Re-engagement Lessons Formative Re-engagement Lessons

Socio-Mathematical Norms Errors are gifts…they promote discussion and learning Making sense is important…not only the answer. Ask questions…until it makes sense. Think with language…use language to think. Use multiple strategies…multiple representations.

Session 1

Math/Number Talks

A Quote from Ruth Parker “I used to think my job was to teach students to see what I see. I no longer believe this. My job is to teach my students to see; and to recognize that no matter what the problem is, we don’t all see things the same way. But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.”

What are Number/Math Talks? Number/Math Talks are a daily ritual with the entire class for the purpose of developing conceptual understanding of and efficiency with numbers, operations and mathematics. (Approximately 10 minutes per day.)

Why Number Talks? Student sense-making is most important to all mathematics teaching and learning. They provide a foundation for both numerical and algebraic reasoning. Students use the nine properties as they explain their thinking. (Properties are the same 3rd grade through calculus.)

The Properties of Operations Associative property of addition (a + b) + c = a + (b + c) Commutative property of addition a + b = b + a Additive identity property a + 0 = 0 + a = a Existence of additive inverses For every a there exists –a so that a + (–a) = (–a) + a = 0 Associative property of multiplication (a x b) x c = a ( b x c) Commutative property of multiplication a x b = b x a Multiplicative identity property of 1 a x 1 = 1 x a = a Existence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a x 1/a = 1/a x a = 1 Distributive property of multiplication over addition a x (b + c) = a x b + a x c

Math/Number Talks Build Numerical Reasoning Article: Number Talks Build Numerical Reasoning by Sherry Parrish

Shared Learning Number off one to five. Read the introduction on pages 198 – 201. Then read your section of the article and the conclusion “Taking the first Steps”. Share your new knowledge with your group.

Dot Talks

Analyzing a MARS Task

Roxie’s Photo 7th Grade Take a few minutes to complete familiarize yourself with the task. Answer all the questions. Discuss your results with a partner. Using the rubric score your own task.

Describe the “Story of the Task” What are the important mathematical ideas of the task? How was the task designed? How did the design contribute to the cognitive demand?

Scoring the Task

Purpose of Scoring Gather data about student thinking to inform and improve instruction. Rubrics designed by international team to reflect shared values and perspectives. Rubrics provide one means of analyzing student work and giving teachers feedback. Scoring consistency allows us to capture data and gain insight into student thinking.

Scoring Principles Different from other scoring systems Points are awarded throughout a task to emphasize varying aspects of doing mathematics “Is there more evidence of understanding or not understanding?” Mathematically equivalent expressions or alternative strategies get full credit. If you need to debate what the student was doing, the explanation was not complete.

Task Design Entry level part - allow access Ramp up - not all parts are equal Recent year’s tasks written to Common Core Standards and Mathematical Practices

Rubrics Embody value judgments and explicit Computation and representation How to tackle an unfamiliar problem Interpret and evaluate solutions Communicate results and reasoning to others Carefully considered evaluation of performance

Scoring Marks √ correct answer or comment x incorrect answer or comment √ft correct answer based upon previous incorrect answer called a follow through ^ correct but incomplete work - no credit ( ) points awarded for partial credit. m.r. student misread the item. Must not lower the demands of the task -1 deduction

Looking at Student Work

Picking Student Work Pick a few key examples from student work that represent: Common strategies Novel approaches Misconceptions

Writing a Re-engagement Lesson

Purpose Formative Assessment Re-engagement Lessons use student work to: Confront misconceptions Provide feedback on student thinking Help students go deeper into the mathematics

Lesson Progression Start with a simple problem Make sense of another person’s strategy Analyze misconceptions and discuss why they don’t make sense. Find out how a strategy could be modified to get the right answer.

Lesson Protocol Three-step Protocol Ask the question and give students individual think time. Share ideas with a partner Whole class discussion.

Types of Re-engagement Lessons Clarifying an idea Comparing strategies Making generalizations about tupes of problems Confront misconceptions Learn a new strategy Model qualities or characteristics of exemplary work

Design a Re-Engagement Lesson Use examples of student work to formulate a question to re-engage all students in the mathematics of the task.

Mathematical Practices Make sense of problems and persevere in solving them …start by explaining the meaning of a problem and looking for entry points to its solution Reason abstractly and quantitatively …make sense of quantities and their relationships to problem situations Construct viable arguments and critique the reasoning of others …understand and use stated assumptions, definitions, and previously established results in constructing arguments Model with mathematics …can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace TIME: 7 minutes (for Slides 44 to 46) INTENT: To understand the Standards for Mathematical Practice as outlined in the CCSS ___________________________________________________________________________________ Have participants locate the CCSS for Mathematics handout and turn to the Standards for Mathematical Practice found on pp.1-2. Briefly discuss each of the Standards for Mathematical Practice. HANDOUTS: CCSS for Mathematics 2011 © CA County Superintendents Educational Services Association California’s Common Core State Standards: Toolkit | Overview

Mathematical Practices Use appropriate tools strategically …consider the available tools when solving a mathematical problem Attend to precision …communicate precisely using clear definitions and calculate accurately and efficiently Look for and make use of structure …look closely to discern a pattern or structure Look for and express regularity in repeated reasoning …notice if calculations are repeated, and look for both general methods and for shortcuts TIME: 7 minutes (CONTINUED for Slides 44 to 46) INTENT: To understand the Standards for Mathematical Practice as outlined in the CCSS ___________________________________________________________________________________ Continued from previous slide. Briefly discuss Standards 5-8. HANDOUTS: CCSS for Mathematics 2011 © CA County Superintendents Educational Services Association California’s Common Core State Standards: Toolkit | Overview

Mathematical Practices Overarching habits of mind of a productive mathematical thinker Reasoning and explaining Modeling and using tools Seeing structure and generalizing 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 1. Make sense of problems and persevere in solving them. 6. Attend to precision. TIME: 5 minutes (CONTINUED for Slides 43 to 47) ___________________________________________________________________________________ William McCallum, one of the writers of the CCSS for Mathematics, has paired the Standards for Mathematical Practice in the following way. Review the slide. When we look at the Standards for Mathematical Practice, we can see how these standards support the kinds of thinking and work described in Quadrants B, C, and D. This organization of the standards helps us consider the implications the CCSS will have on instruction. In addition to teaching content, we need to plan instruction to help students make sense of problems and persevere in solving them—a challenge we’ve always been faced with in mathematics, but now it is a standard we need to help students achieve. In our lesson plans, we need to create opportunities for students to engage in reasoning and explaining or using multiple representations and tools to model problems. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning adapted from McCallum (2011) Standards for Mathematical Practice California’s Common Core State Standards: Toolkit | Instruction, K-5 | ELA and Mathematics

Sharing Re-engagement Lessons

Sharing Lessons Please share your lesson with the group.

Session 2

Math/Number Talks

Guiding Principles for Number Talks From Ruth Parker All students have mathematical ideas worth listening to and our job as teachers is to help students learn to develop and express these ideas clearly. Through our questions we seek to understand students’ thinking. We encourage students to explain their thinking conceptually rather than procedurally. Mistakes provide opportunities to look at ideas that might not otherwise be considered. While efficiency is a goal, we recognize that whether or not a strategy is efficient lies in the thinking and understanding of each individual learner.

Guiding Principles for Number Talks From Ruth Parker We seek to create a learning environment where all students feel safe sharing their mathematical ideas. One of our most important goals is to help students develop social and mathematical agency. Mathematical understandings develop over time. Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics. We value and encourage a diversity of ideas.

Making Tens 9 + 1 9 + 3 + 1 9 + 5 + 1 5 + 5 + 8 3 + 4 + 6 4 + 5 + 6 + 5

Doubles or Near Doubles 3 + 3 3 + 4 3 + 2 12 + 12 13 + 13 12 + 11 12 + 13

Making Landmarks or Friendly Numbers 10 + 2 9 + 2 9 + 5 9 + 8 9 + 9 40 + 4 38 + 2 + 2 38 + 4 38 + 13 38 + 27

Making Landmark or Friendly Numbers 4 x 25 4 x 200 4 x 250 4 x 249 10 x 30 2 x 30 12 x 29

Keeping a Constant Distance 14 – 10 13 – 9 14 – 7 13 - 6 90 – 45 89 – 44 91 – 46 98 – 52

Multiplication Across the Grades From Ruth Parker Four strategies for multiplication Factor x Factor = Product Break a factor into two or more addends. Factor a factor Round a factor and adjust. Halving and doubling.

Break a Factor into Two or More Addends 12 x 16 12 x 16 = 12 x ( 10 + 6) = (12 x 10) + (12 x 6) = 120 + 72 = 192

Factor a Factor 12 x 16 = (12 x 4) x (2 x 2) = 48 x (2 x 2) = (48 x 2) x 2 = 96 x 2 = 192

Round a Factor and Adjust 12 x 16 12 x 20 = 240 12 x 4 = 48 240 – 40 = 200 200 – 8 = 192 16 = 20 - 4 40 + 8

Halving and Doubling 12 x 16 = 24 x 8 = 48 x 4 = 96 x 2 = 192

Math/Number Talk 14 x 12 =

Re-Engagement Lessons Classroom Challenges (from the MAP Site) 6th, 7th and 8th Grade

Mathematics Assessment Project http://map.mathshell.org

Instructions To get started, click on the Lessons or Tasks tab at the tope of the screen. Then use the drop-down menu to choose the grade level.

Browse by Lesson or Task

Browse by Standard

What are MAP Classroom Challenges? MAP Classroom Challenges (CCs), also know as formative assessment lessons (FALs) include: Mathematical investigations Lessons Tasks Assessments Cooperative group collaborations

Why Use Classroom Challenges? Allows students to demonstrate their prior understandings and abilities in employing the math practices Involves students resolving their own difficulties and misconceptions Results in secure long-term learning Reduces the need for re-teaching

Two Types of CCs Concept Development Lessons Problem Solving Lessons Reveal students’ prior knowledge Develop students’ understanding of important mathematical ideas Connect concepts to other mathematical knowledge Problem Solving Lessons Assess then develop students’ capacity to apply their mathematics flexibly to non-routine unstructured problems

Structure of the CCs Concept Development Problem Solving Pre-Assessment to reveal capabilities/limitations in problem solving Pre-Assessment to identify common issues Lesson is done in small groups Lesson designed to expose different ideas After the lesson students work alone to improve their individual solutions Post-Assessment Problem Solving

Genres of Activities Used in Concept Development Lessons The main activities in the concept lessons are built around the following four genres. Classifying mathematical objects Interpreting multiple representations Evaluating mathematical statement Exploring the structure of problems

Top Ten Reasons for Using CCs from teachers who have used them. Students’ understanding of math expands and deepens. Enables teachers to implement the Common Core Standards and Practices Engages and tests students of all abilities. Demands that students talk and write about mathematics. Increases the excitement level in the classroom.

Top Ten Reasons for Using CCs from teachers who have used them. Stress conceptual understanding Helps teachers shift from teacher-centered to student directed classrooms. Allows teachers to hear and see their student in new way Expertly designed and ready to use Enables teachers implement active listening, questioning and facilitating small groups.

Interpreting Algebraic Expressions Matching expressions, words, tables and areas. Choose and expression, words, table or area and the find the other three representations. If you cannot find a matching expression, words, table, or area, then you should write your own. Justify your matches and explain your thinking.

Interpreting Algebraic Expressions Translating between words, symbols, tables, and area representations of algebraic expressions. Do students Recognize the order of algebraic operations? Recognize equivalent expressions? Understand the distributive laws of multiplication and division over addition (expansion of parentheses)?

Interpreting Algebraic Expressions

Session 3

Math Talk Jigsaw Overview In groups of 3 plan a Number Talk Present your Number Talk to another group. Return to your preparation group to debrief presentation and reflect.

In Your Like Groups Plan a multiplication number talk that you will use in your classroom. Brainstorm with your group as many different strategies and solution paths you can come up with for your particular problem. Share ideas about how you will record “student” strategies and solutions. Share ideas about how to handle potential pitfalls in the presentation and recording of your particular problem.

In Groups of Four Present your Math Talk to the group.

Like Group/Whole Group Debrief Debrief and reflect on your math talk: What went well? What surprised you? What changes do you want to make?

Chalk Talk

Chalk Talk This is a completely silent activity Ideas are expressed by writing on the chart paper. Comments on what others have written must be added to the chart paper. What did you learn today? What is most important to share? How can you share? What materials will you need?

Chalk Talk So what? Now what? How often? Every day? Three times a week? What are the norms? Who will create them? Grade level teams? Cross grade levels? Individual teachers? What protocols would we need?

Wrap-up