Brentwood • Byron • LibertyCoaching Institute

October 7th – October 9th and Collaborative Tasks

Silicon Valley Mathematics Initiativewww.svmimac.org

6th – 8th GradeBreakout Sessions

April Cherrington

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.

4

Session 1

5

Math/Number Talks

A Quote from Ruth Parker

“I used to think my job was to teach students tosee what I see. I no longer believe this. My jobis to teach my students to see; and to recognizethat no matter what the problem is, we don’tall see things the same way. But when weexamine our different ways of seeing, and lookfor the relationships involved, everyone seesmore 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

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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?

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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

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Looking at Student Work

Picking Student Work

Pick a few key examples from student work that represent:

• Common strategies

• Novel approaches

• Misconceptions

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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

1. Start with a simple problem2. Make sense of another person’s strategy3. Analyze misconceptions and discuss why

they don’t make sense.4. Find out how a strategy could be

modified to get the right answer.

Lesson Protocol

Three-step Protocol1. Ask the question and give students individual

think time.2. Share ideas with a partner3. 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 Practices1. 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

2. Reason abstractly and quantitatively…make sense of quantities and their relationships to problem situations

3. Construct viable arguments and critique the reasoning of others…understand and use stated assumptions, definitions, and previously established results in constructing arguments

4. Model with mathematics…can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace

2011 © CA County Superintendents Educational Services Association30

Mathematical Practices5. Use appropriate tools strategically

…consider the available tools when solving a mathematical problem

6. Attend to precision…communicate precisely using clear definitions and calculate accurately and efficiently

7. Look for and make use of structure…look closely to discern a pattern or structure

8. Look for and express regularity in repeated reasoning…notice if calculations are repeated, and look for both general methods and for shortcuts

2011 © CA County Superintendents Educational Services Association31

Mathematical Practices

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adapted from McCallum (2011)Standards for Mathematical Practice

Overarching habits of mind of a productive mathematical thinker

Reasoning and explaining

Modeling and using tools

Seeing structure and generalizing

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2. Reason abstractly and quantitatively.3. Construct viable arguments and

critique the reasoning of others.

4. Model with mathematics.5. Use appropriate tools strategically.

7. Look for and make use of structure.8. Look for and express regularity in

repeated reasoning

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Sharing Re-engagement Lessons

Sharing Lessons

Please share your lesson with the group.

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Session 2

36

Math/Number Talks

Guiding Principles for Number TalksFrom Ruth Parker

1. 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.

2. Through our questions we seek to understand students’ thinking.

3. We encourage students to explain their thinking conceptually rather than procedurally.

4. Mistakes provide opportunities to look at ideas that might not otherwise be considered.

5. 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 TalksFrom Ruth Parker

6. We seek to create a learning environment where all students feel safe sharing their mathematical ideas.

7. One of our most important goals is to help students develop social and mathematical agency.

8. Mathematical understandings develop over time.9. Confusion and struggle are natural, necessary, and

even desirable parts of learning mathematics.10. We value and encourage a diversity of ideas.

Making Tens

9 + 19 + 3 + 19 + 5 + 1

5 + 5 + 83 + 4 + 6

4 + 5 + 6 + 5

Doubles or Near Doubles

3 + 33 + 43 + 2

12 + 1213 + 1312 + 1112 + 13

Making Landmarks or Friendly Numbers

10 + 29 + 29 + 59 + 89 + 9

40 + 438 + 2 + 2

38 + 438 + 1338 + 27

Making Landmark or Friendly Numbers

4 x 254 x 2004 x 2504 x 249

10 x 3010 x 302 x 30

12 x 29

Keeping a Constant Distance

14 – 1013 – 914 – 713 - 6

90 – 4589 – 4491 – 4698 – 52

Multiplication Across the GradesFrom Ruth Parker

Four strategies for multiplicationFactor x Factor = Product

1. Break a factor into two or more addends.2. Factor a factor3. Round a factor and adjust.4. 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 1612 x 20 = 24012 x 4 = 48

240 – 40 = 200200 – 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 =

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Re-Engagement LessonsClassroom 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– 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 • Pre-Assessment to identify

common issues

• Lesson designed to expose different ideas

• Post-Assessment

Problem Solving

• Pre-Assessment to reveal capabilities/limitations in problem solving

• Lesson is done in small groups

• After the lesson students work alone to improve their individual solutions

Genres of Activities Used in Concept Development Lessons

The main activities in the concept lessons are built around the following four genres.1. Classifying mathematical objects2. Interpreting multiple representations3. Evaluating mathematical statement4. Exploring the structure of problems

Top Ten Reasons for Using CCsfrom teachers who have used them.

1. Students’ understanding of math expands and deepens.

2. Enables teachers to implement the Common Core Standards and Practices

3. Engages and tests students of all abilities.4. Demands that students talk and write about

mathematics.5. Increases the excitement level in the

classroom.

Top Ten Reasons for Using CCsfrom teachers who have used them.

6. Stress conceptual understanding7. Helps teachers shift from teacher-centered to

student directed classrooms.8. Allows teachers to hear and see their student

in new way 9. Expertly designed and ready to use10.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

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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?

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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.

Chalk TalkSo 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?

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Wrap-up