1. 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

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

3. 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

6. 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.”

7. 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.)

8. 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.)

9. 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 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

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

11. 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.

12. Dot Talks

13. Analyzing a MARS Task

14. 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.

15. 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?

16. Scoring the Task

17. 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.

18. 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.

19. 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

20. 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

21. 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

22. Looking at Student Work

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

24. Writing a Re-engagement Lesson

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

26. 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.

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

28. 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

29. 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.

30. 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

31. 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

32. 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

33. Sharing Re-engagement Lessons

34. Sharing Lessons
Please share your lesson with the group.

35. Session 2

36. Math/Number Talks

37. 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.

38. 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.

39. Making Tens
9 + 1

40. Doubles or Near Doubles
3 + 3
3 + 4
3 + 2

41. Making Landmarks or Friendly Numbers

42. Making Landmark or Friendly Numbers
4 x 25 4 x x x 249
10 x 30 2 x x 29

43. Keeping a Constant Distance
14 – – 9 14 –
90 – – – – 52

44. 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.

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

46. 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

47. Round a Factor and Adjust
12 x x 20 = x 4 = – 40 = – 8 = 192
16 = 40
+ 8

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

49. Math/Number Talk
14 x 12 =

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

51. Mathematics Assessment Project

52. 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.

53. Browse by Lesson or Task

54. Browse by Standard

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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.

61. 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.

62. 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.

63. 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)?

64. Interpreting Algebraic Expressions

65. Session 3

66. 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.

67. 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.

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

69. 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?

70. Chalk Talk

71. 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?

72. 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?

73. Wrap-up