1. Balanced Math Framework
August 15, 2013
8:30-3:30
11:30-12:30 lunch

2. Getting to Know You... Math Style
Grab a bingo card from the middle of your table
Circulate the room searching for teachers who can "sign" a box on your bingo card (one signature per box please)
Slide minutes - 8:30-8:40
Each teacher will need a bingo card and a pen/pencil. They will move around the room and attempt to fill up all the boxes on their bingo card by finding someone who that box applies to. The rules: (1) each person you talk to may only sign your sheet once, and (2) to win, you must get signatures in all of the boxes.

3. Math Workshop Readers and Math Writers = Workshop Workshop Framework
Discuss how Readers and Writers workshop philosophy or framework parallels with math workshop. Flexible components based on student needs .

4. Side 3 - 5 minutes - 8:40-8:45 (See handout)
Side minutes - 8:40-8:45 (See handout). Weekly math instruction should incorporate all these components, but not necessarily every component everyday.
District expectations include daily mental math and math review, conceptual understanding. Fluency should occur at least three times a week.

5. Math Review is
Time to reinforce a previously taught concept
Formative and based on daily student understanding
Work that is de-briefed and discussed
Used to guide instruction
3 to 6 review problems (based on grade level)
An opportunity to circulate and observe common misconceptions or understandings
Slides 4 & minutes - 8:45-8:50 (Balanced Math component #1)The resource in your new Expression Series has a component called Homework and Remembering that has examples of problems that can be used to help build your math review. This is not a resource you will just copy and give to students. Remember Math Review is based on need, so a pre-made resource will never meet the individual needs of your classroom.

6. Math Review is
Not time to teach a new concept or trick the students
Not pre-printed or planned by yearlong or unit objectives
Not work completed without discussion
Not used as a grade or graded by others
Not more than six problems
Not busy work

7. Math Review and Mental Math
Slides minutes - 8:50-9:10 - Teachers will work through these problems. Circulate and choose a teacher to demonstrate a misconception or error to highlight as well as someone who has shown a strategy you would like for all to see. Those people will come to the front and share their work. The "class" will discuss the problem. Students will be looking at their own work and self assessing how they did (mark up their paper to show where they messed up and/or if they were correct). Also highlight the math vocabulary. MENTAL MATH PORTION: For the mental math part the presenter will read aloud three mental math strings. Read them slowly so that the participants have time to mentally solve in their head (No writing or scratch paper). Participants will write their answer on their paper. Ask the participants for each answer all at once and choose one volunteer to verbally describe his/her mental math strategy.
Mental Math Strings to use:
1.) Find the difference of 12 and 9. Increase by 7. Multiple by 10.
2.) Find the sum of 5 and 9. Double it. Divide by 7.
3.) Take the number of sides of a cube and multiple by 9. Subtract 4. Divide 2.
The context of the problem will remain the same for two week. Only the numbers may change. On the final Friday the math review will serve as an summative assessment of the two week cycle. This assessment may serve as a guideline for the next math review cycle.

8. This is another template that could be used
This is another template that could be used. It is not necessary to cover all domains each day. A template will be included in your curriculum. One idea might be to keep your desired template on a Powerpoint slide. Each day you just plug in your problems. When it comes time to build your assessment for day 10, you just use the previous 9 days for a resource. Hand out this example. You do not have the next two examples. This is attached at the course level.

11. Problem Solving - happens daily in the classroom.
Slides :10-9:15 - (Balanced Math Component #2) We won't go into the poster method of problem solving today but we will discuss this component of Balanced Math later in time.

12. Conceptual Understanding
This is where you teach your curriculum. You will use Math Expressions, Glencoe and DMI experiences as a resource.
(Balanced Math Component #3) This is the meat of your math time. When beginning to plan out your math lessons using the Expression/Glencoe (6th) Series, if you read and understand the first 5 lessons you will have a conceptual understanding of the mathematical thinking that is involved and what you want your students to take from those lessons. Then each night if you read one more lesson, you will always be 5 days out in your planning and thinking. Karen Fuson (the author of Math Expressions) has said her material is designed for teachers to read five days out.

13. Slides :15 - 9:25 This slide is an introduction to the Standards of Mathematical Practices, they will be more visible on slide 16

14. Read the info on the slide: The Standards for Mathematical Practice (referred to as the SMP in the curriculum) are the processes in which students DO the math. Much like the old blue placemat, these are the processes for math only according to the CAS. They are part of every domain in math grades K-12. We've always had "Content Standards"....the Standards for Mathematical Practice are the driving force behind the instructional change.

15. Read the quote and let teachers process the info.

16. Standards for Mathematical Practice
Mathematically Proficient Students...
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeating reasoning.
Attached to curriculum and handed out at rollout meetings was a bulleted list that provides more specific suggestions on making this practice standard come alive in your classroom. Remember, #4 is the student doing the modeling not the teacher.

17. The Standards for Mathematical Practice
Take a moment to examine the first three words of each of the 8 mathematical practices... what do you notice?
Mathematically Proficient Students...
Slide :25-9:40: - See handout of the bulleted Mathematical Practices to do this activity.

18. This slide summarizes the thought from the previous slide "Mathematically Proficient Students...": The following terms above/ beside each colored box are used throughout the math resource "Math Expressions". Notice that Math Sense Making involves persevering in problem solving and attending to precision, Math Explaining requires students to reason abstractly and quantitatively and to construct viable arguments while critiquing the reasoning of others, Math Drawings involve modeling math and using the appropriate tools, and Math Structure groups looking for and using math structures and repeated reasoning. One or more of these practice standards are used in every lesson or activity.

19. The Standards for [Student] Mathematical Practice
What are the verbs that illustrate the student actions of each mathematical practice?
Now that we have looked at the big ideas of the SMP, let's look at what this really means for what students will actually be doing. Jigsaw activity: Assign each table/group one of the SMP. Explain that teachers should now be looking at the bulleted points below each numbered SMP that they were assigned.. They should highlight, circle, or underline the verbs. Give teachers about 5 min. to do this. When finished ask teachers to briefly discuss their thoughts about each standard with the whole group. The next slide is an example of SMP #3.

20. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others
Mathematically proficient students:
• understand and use stated assumptions, definitions, and previously established results in constructing arguments.
• make conjectures and build a logical progression of statements to explore the truth of their conjectures.
• analyze situations by breaking them into cases, and can recognize and use counterexamples.
• justify their conclusions, communicate them to others, and respond to the arguments of others.
• reason inductively about data, making plausible arguments that take into account the context from which the data arose.
• compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is.
• construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
• determine domains to which an argument applies.
• listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Link to Blooms

21. In the SJSD curriculum...
In the SJSD each learning target has identified the SMP students will be using to meet that standard (formerly referred to as an objective).

22. Slides :40 - 9:55 Buttons Task: Handout activity - similar to an activity/task students will be doing. This task was originally tagged as a 5th grade task, but with CAS it is now a 4th grade task and is attached to the SJSD curriculum. Don't actually do the task. Just read through and discuss what standards of mathematical practice you think will take place. Then we will watch the video and they can see it come to life.

26. Standards for Mathematical Practice
Buttons Task
Tell teachers as they are watching to identify which of the 8 standards of mathematical practices they are observing. Video is six minutes.

27. Standards for [Student] Mathematical Practices
• "Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking."
~ Stein, Smith, Henningsen, & Silver, 2000
• "The level and kind of thinking in which students engage determines what they will learn."
~ Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human
1997
Slides :55-10:05 Comparing two tasks Goal for the Standards of Mathematical Practices is to make these standards come alive in our classrooms. Just having the numbers aligned to our lessons is not enough. This is a transition slide. The others are shared reading.

28. Comparing Two Mathematical Tasks
Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?
~ Stein, Smith, Henningsen, & Silver, 2000, p. 1
This is typical of what we see now. This is what we think of as a CR question on the MAP test

29. Comparing Two Mathematical Tasks
Ms. Brown's class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits.
1. If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?
2. How long would each of the sides of the pen be if they had only 16 feet of fencing?
3. How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who read it will understand it.
~ Stein, Smith, Henningsen, & Silver, 2000, p.2
This is more typical of what we could see now. Ask who in the room got to watch the math pilot What did they see? This slide also sets up to discuss the four quadrants of Rigor and Relevance. This would be a quad 4.

30. Comparing Two Mathematical Tasks
Discuss:
How are Martha's Carpeting Task and the
Fencing Task the same and how are they
different?

31. Comparing Two Mathematical Tasks
Lower-Level Tasks Higher-Level Tasks

32. Reflection
My definition of a good teacher has changed from "one who explains things so well that students understand" to "one who gets students to explain things so well that they can be understood."
(Steven C. Reinhart, "Never say anything a kid can say!" Mathematics Teaching in the Middle School 5, 8 [2000]: 478)
What do you think Steve means by his definition of a good teacher?
Imagine Steve’s classroom. What do you think you might see and hear?
In your own learning, what experiences have you had where talking and listening were used to support your understanding? What was it like?

33. Richard Schaar
What I learned in school may be growing increasingly obsolete today, but how I learned to learn is what helps me keep up with the world around me. I have the study of mathematics to thank for that.

34. 10:10-10:25

35. Rigor and Relevance

36. Rigor & Relevance Framework
Relevance makes RIGOR possible, but only when trusting and respectful relationships among students, teachers, and staff are embedded in instruction. Relationships nurture both rigor and relevance.
10:25-10:27 minutes - Read slide

37. Rigor is...
10:27-10:30 - Have teachers write their own definition of Rigor. Keep this definition close by because we will revisit it in a few minutes.

38. Tips for Using Rigor, Relevance and Relationships.
Article:
Tips for Using Rigor, Relevance and Relationships.
10:30-10:45 - Teachers should read the article "Tips for Using Rigor, Relevance and Relationships" and in the margin summarize in 10 words or less what they think the author is saying per section. There are 2 sections. Table share of margin summaries. Group share of take aways from the article.
Pose the question: "Has anyone's definition of rigor changed after reading this article or from our discussion?"

39. Rigor is...
Work that requires students to work at high levels of Bloom's Taxonomy combined with application to the real world.
10:45-:10:55 - Slides
Read out loud

40. 3 Misconceptions of Rigor
•MORE – does not mean more rigorous.
•DIFFICULT – increased difficulty does not mean increased rigor.
•RIGID – “all assignments are due by… no exception.”
Example for "Difficult" could be without support of our students. We can still support our students through facilitation of lessons/mathematical thinking and it will be rigorous.

41. RIGOR

42. The thinking continuum on the previous slide is defined by Blooms
The thinking continuum on the previous slide is defined by Blooms. This is the vertical axis on the rigor and relevance framework. The Bloom's level is defined in the unwrapping of the new ELA and math curriculum. This is the REVISED Bloom's. Handout provided.

43. Why do I need to know this?
Relevance
Why do I need to know this?
This is the horizontal axis on the Rigor and Relevance Framework. The action continuum.

44. Misconceptions of Relevance
•COOL – relevance doesn’t exclusively mean what the students do for “fun”
•EXCLUSIVE – relevance without rigor does not ensure success.

45. Relevance

46. Application Model 1. Knowledge in one discipline
2. Application within discipline
3. Application across disciplines
4. Application to real-world predictable situations
5. Application to real-world unpredictable situations
Presenter will share the example listed below for each level of the application model
1.Label foods by nutritional groups
2.Rank foods by nutritional value
3.Make Cost comparisons of different foods considering nutritional values
4.Develop a nutritional plan for a person with a health problem affected food intake
5.Devise a sound nutritional plan for a group of 3-year-olds who are picky eaters

47. Putting it all together
10:55- 11:00 Slides 47-50

48. The quad is determined for you in your curriculum.
Handout provided. Note that the 4 quadrants are not the same in size. This difference represents where students should the majority of their instructional time. All quads have various levels of rigor/application.
A – Students gather and store bits of knowledge and information – REMEMBER and UNDERSTAND
B – Use acquired knowledge to solve problems, design solutions, ad complete work – unpredictable situations
C – Extend and refine their acquired knowledge to be able to use that knowledge automatically and routinely to analyze, solve, and create unique solutions.
D – Think in complex ways and apply knowledge they’ve acquired. Even in perplexing unknowns, students use knowledge to create solutions.
The quad is determined for you in your curriculum.

49. Read slide - Note that in the upper two quadrants the students are doing more than just completing assigned work, they are thinking and applying their thinking to their task. Goal is to decide which quadrant best aligns with the standard/target you are creating a lesson for. There are times quad A is appropriate but we want to push our lessons out of quad a to other quads.

50. This slides shows how Rigor and Relevance aligns with Bloom's Taxonomy.

51. Rigor and Relevance Card Sort
Activity
Rigor and Relevance Card Sort
11:00-11:20 - Teachers should use a copy of the revised Bloom's Taxonomy and the Application Model Decision Tree to sort a group cards into the different quadrants. See handouts that go with activity. Share the answers at the end of the activity.

52. Six Questions All Students Must Be Able to Answer
When seeking rigor, relevance, and relationships, all students should be able to answer the following questions:
1. What is the purpose of this lesson?
2. Why is this important to learn?
3. In what ways are you challenged to think in this
lesson?
4. How will you apply, assess, or communicate what
you've learned?
5. Do you know how good your work is and how you can
improve it?
6. Do you feel respected by other students in this class?
Slides :20-11:30 This is ending the discussion of Component #3 of the balanced math framework (Conceptual Understanding).

53. Mastery of Math Facts
After students have reached conceptual understanding, the following fluencies are required by the CAS:
K K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2 2.OA.2 Add/subtract within 20 (know single digit products from memory)
2.NBT.5 Add/subtract within 100
A.7 Multiply/divide within 100 (know single-digit products from memory).
3.NBT.2 Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
NBT-5 Multi-digit multiplication
NS.2,3 Multi-digit division Multi-digit decimal operations
(Balanced Math Component #4) It is important in K and 1 that we don't drill and kill for fact memorization without conceptual understanding. Conceptual understanding has to happen first before automaticity happens for true fluency. A student needs to understand that 5 has a 4, 3, 2, 1 in it and understand that fact families leading up to 5 can help solve problems with five. These fluencies do not have to be mental (unless they say "from memory"). A student just needs to be able to show that they have a math strategy to figure out the answer quickly. Reflex is your fact fluency resource to monitor your student's mastery of facts.
For example: if a kindergartner is posed this problem = and says 3 and counts up 2 to 5... that is exhibiting fluency.
Another example if a 3rd grader has the problem 6 x 6 = and says 6 x 5 = = 36, that is still fluency. Fluency does not have to mean rote memorization. Just a quick recall using a number sense strategy.

54. Common Formative Assessment
Kindergarten and First Grade Mathematics Interviews
Math Fact Fluency - Reflex
Conference Notes - anecdotal records and Math Reasoning Inventory
Performance Tasks
Mathematics Predictive Exams
Math Review and Mental Math
(Balanced Math Component #5). Defined in the Balanced Mathematics Framework attached at the course level in curriculum.

55. For Session 1: Please read Casebook pages 13-28 Cases 3, 4, 5
One assignment should be done each quarter.
Distribute Third Homework, p. 75 facilitator's guide
This work will be followed up on in JEPD.

56. Lunch
11: :30

57. Developing Mathematical Ideas
Have the tables set up with name cards where you have placed people. We want to use the strategy that Virginia used where we can create groupings of teachers to work together in different sessions.
Hang up a blank poster labeled Math Vocabulary - add tier 3 vocabulary as it comes up throughout the 2 days.
August 15 & 16, 2013

58. DMI is about developing YOUR mathematical understanding
“If our goal is to create mathematically powerful children then we must also
create mathematically powerful teachers.”
--Lance Menster
Introduction (Slides 1 to 5) - 5 minutes (12: :35)

59. What is DMI?
Developing Mathematical Ideas (DMI) is a professional development curriculum presented through a series of seminars.
The premise of the DMI materials
is that the art of teaching involves
helping students move from where
they are into the content to be
learned.
Make a point to let teachers know that this about their adult learning.

60. DMI Premises
DMI seminars bring together teachers from kindergarten through middle grades to:
• learn mathematics content
• learn to recognize key mathematical ideas
with which their students are grappling
• learn how core mathematical ideas develop
across the grades
• learn how to continue learning about children
and mathematics

61. DMI is a Process • This year we are working through the first
module: Building a System of Tens
• Today and tomorrow we are working in the
first 3 sessions
Session 1: Analyzing Addition Strategies
Session 2: Place Value and Multiplication
Session 3: The Mathematics of Algorithms
DMI is made of 7 modules. This year our focus is Module one - building a system of tens. It comprises 8 sessions.

62. Session One: Building a System of Tens
Student's Addition and Subtraction Strategies
Facilitator Guide pages

63. Mathematical Goals for Session One
I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations.
I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works.
I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.).
12: :40

64. Mental Math
20 minutes - Slides (12:40 - 1:00). (Facilitator Guide page ). Share out the activity on ELMO/Doc camera instead of chart paper so that everyone in the room can see. Keep asking for different strategies. "Did anyone use a different strategy?" Ask participants how each of these strategies takes advantage of the tens structure of the number system and what similarities and differences they note among them.

65. Mental Math
83-56
After completing each of the mental math activities, bring discussion back to the importance of the base ten structure and how it is demonstrated in each example.

66. Mathematical Goals for Session One
I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations.
I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works.
I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.).
5 minutes (1:00 - 1:05). After completing the activity, briefly reflect on how we have covered these goals so far.

67. Second Grade Strategies
25 minutes (slides 67-68) - 1:05 to 1:30. (See page 46 Maxine also Facilitator Guide page 22, for summary) Before starting the DVD, solicit one or two approaches to so that participant methods are on display. Have teachers take notes over the DVD and the strategies the students are using to solve the problem. Stop the DVD after each student to discuss the student's strategy before showing the next student. (Note: The DVD is finicky. Practice with it first. There are no "chapters" for easy navigating. It is easiest to pause by hitting the space bar on the computer. Have one person be in charge of the computer when using the DVDs. Maybe have two computers and switch out the dongle??)

68. Seventh Grade Strategies
(See Maxine page 46) Before showing the 7th grade segment, ask participants to think about and record some of their approaches. You might suggest that they try an approach that is new to them. Then show the DVD clip.
Stop after the kids give what they know about the answer and then after each strategy. Bring out how asking what you know about the answer helps children to develop whether an answer is reasonable. If you are running short on time you can stop less, but be sure to stop after the ones with the more challenging math.

69. Mathematical Goals for Session One
I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations.
I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works.
I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.).
5 minutes (1:30 - 1:35) After completing the activity, briefly reflect on how we have covered these goals so far.

70. Break
15 minutes (1:35 - 1:50)

71. Chapter 1 Case Discussion
In your group, examine Focus Questions 3, 4, and 5. Use any manipulatives or chart paper you need to work through these questions.
50 minutes 1: :40 for slides (25 for small group discussion and 25 for large group discussion). Hand out Focus Questions: Chapter 1 page (31) in the facilitator's book.

72. Small Group Discussion
Possible discussion questions that can apply to both Small Groups and Whole Groups:
1. Stories are tools for reasoning.
2. Where is base 10 represented in this strategy?
3. What actions can you take to continue the learning? What is the next step with this student?
4. How are these strategies alike? How are they different?
5. There are 2 different ways to think about subtraction.
6. Can this strategy be represented as a model? # line? story context?
7. I hear you saying WHEN IN DOUBT..SHUT UP AND MARINATE:) (See Facilitator Guide page )

73. Whole Group
Discussion
(See Facilitator Guide page 24)

74. Mathematical Goals for Session One
I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations.
I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works.
I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.).
5 minutes (2:40 - 2:45) After completing the activity, briefly reflect on how we have covered these goals so far.

75. Math Activity: Close to 100 Game
The object of the game is to create two 2-digit numbers whose sum is as close to 100 as possible. Each game has five rounds. At the end of five rounds the player with the lowest total score wins.
30 minutes (2:45 - 3:15). Each school responsible for making decks ( 1 set per pair). Handouts p Be sure to discuss the strategies that the teachers are using. It may be a good idea to pause after the teachers have been playing for a while to have them share strategies instead of waiting until the end. (See Facilitator Guide page )

76. Mathematical Goals for Session One
I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations.
I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works.
I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.).
5 minutes (3:15 - 3:20) After completing the activity, briefly reflect on how we have covered these goals so far.

77. For Session 2, please be sure to read:
Case studies 6, 7, & 10
Read case studies 6, 7, 10 for session 2. (Questions 1, 2, 4,5) (Casebook page and )

78. Exit cards
1. What mathematical ideas did this session highlight for you?
2. What was this session like for you as a
learner?
3. What burning questions do you have about
this session?
Teachers can leave once they turn in their cards. (Facilitator Guide page 27)

79. Session Two: Building a System of Tens
The Base Ten Structure of Numbers
August 16, 2013
Have the tables set up with name cards where you have placed people. We want to use the strategy that Virginia used where we can create groupings of teachers to work together in different sessions.
(Facilitator's Guide page )

80. Mathematical Goal for Session Two
The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right.
8:30-8: minutes

81. Math Activity 5 minutes - 8:35-8:40
Begin by asking the group for examples of ways they can represent multiplication. Make a poster of what is offered. Suggestions might include making equal-sized groups, drawing rectangles, drawing arrays, or devising story contexts that involve equal groups. Then have each group start their own work. Use handout on page 73.

82. Small Group: Representing Multiplication
25 minutes - 8:40-9:00 (Facilitator Guide small group activity page 67; #1 on page 73).
In this math activity, making models for problem 1 is particularly important. If a group is confused about extending the base 10 models beyond the thousand block, hold up the 1,000 cube and ask, "What would ten of these look like?" As groups work on problems 2, 3, and 4, ask questions to help participants describe the connections between the quantity a number represents and the way it is written. Participants should articulate their own explanations for why multiplying by 10 does not change the sequence of digits. As you interact with the small groups take notes of the representations they use and chose 2 or 3 that you want to include in the whole-group discussion about the rule for multiplying by 10. It will be useful to have base 10 models, story context, and numerical arguments to compare.
Remember to encourage the teachers look at the shape of each number (shapes may need to be rotated for teachers to get a good mental image).

83. Whole-group Discussion: Sharing Representations
20 minutes 9:00-9:20 (Facilitator Guide page )
Two main ideas for the whole-group discussion:
• Sharing representations for multiplying by 10, 100, and other powers of ten.
• Explaining why the rule "add a zero at the end of a number" works for multiplying by 10.

84. Mathematical Goal for Session Two
The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right.
5 minutes - 9:20-9:25 - Ask teachers to think about how our new thinking meets our goal.

85. DVD: Interview with Three Students
25 minutes - 9:25-9:50 (show all videos) (see Facilitator Guide page ; DVD summary )
Presenter:
"Possible Questions: Front load these questions before you show the videos
First I want us to concentrate on what we are learning about each student's thinking. After we have collected ideas about all three students, we will turn to the interviewers."
"What did you notice about the questions the interviewers asked?"
What does the student know?
What does the student still need to learn?

86. Mathematical Goal for Session Two
The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right.
5 minutes- 9:50-9:55 - Ask teachers to think about how our new thinking meets our goal.

87. Break
15 minutes - 9:50-10:05

88. Case Discussion Think about:
1. What is right about the student's thinking?
2. Where has the student's thinking gone awry?
Slide minutes - 10:05-11:00 - (Case Discussions page 69 in facilitators guide) (Casebook page and cases 6, 7, 10)
(Presenter notes: Through questions 1 and 2, participants work to understand both what is correct in a student's thinking and where their thinking went awry. Questions 4 and 5 examine the number line. As you interact with the small groups, take note of the ways participants use the number line in question 4 and decide which of those representations would be productive for the whole group. The whole-group discussion should illustrate the variety of methods your participants used.)

89. Small-Group: Ideas about the Number System
30 minutes (See Facilitator Guide page 70)
Distribute page 74 facilitator guide (Focus questions)
Assign Questions 1, 2, 4, and 5 for cases 6, 7, and 10.
As you look through the case studies, work to understand both what is correct in a student's thinking within the base ten structure, and where the thinking went awry.
Consider how the different representations would help children learn: e.g. Number lines versus 100s charts. Think about how the representation can create or help a misconception. (Maxine's Journal page 90 Line 500)

90. Whole-Group: Number Lines
25 minutes (See Facilitator Guide page )
Ask participants what surprised them or what they learned from creating their own number lines or examining those of the students.
Make sure to highlight that this type of teaching is not about preventing students from making mistakes but rather to consider what the work of students reveals about their understanding. (Maxine's Journal, pps 91-93, lines ).
Our discussion about how Shaquille and Chris wasn't centered around their mistakes, but instead centered around what the student work reveals their understanding. (Maxine Journal page 92 line 600)

91. Mathematical Goal for Session Two
The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right.
5 minutes - 11:00-11:05 - Ask teachers to think about how our new thinking meets our goal.

92. During your working lunch please be sure to read Case 14
(pages Casebook).

93. Lunch
1 hours (11: :05)

94. Session Three: Building a System of Tens
Making Sense of Addition and Subtraction Algorithms
Facilitator Guide page

95. Mathematical Goals for Sessions Three
• Extend students’ knowledge of place value (ones, tens, hundreds) to solving addition and subtraction problems efficiently.
• Understand how place value underlies the traditional algorithms for addition and subtraction.
5 minutes (12: :10)

96. Whole Group: Addition and Subtraction Strategies
Investigating addition strategies:
• Creating Verbal Descriptions
• Visual Representations
• Story Context
In this session you do whole group before small group (practice together before they do in their own groups)
45 minutes (12: :55) (Page 105 & Page 106 gives an example of a verbal description, a representation, and a story context facilitator's Guide)
In this activity, participants will examine various strategies for addition and subtraction of two-digit numbers to determine what mathematical ideas each strategy calls upon.
Definition: Algorithm is a set of steps that accomplishes a particular task.
Ask for ONLY a verbal description of the different addition strategies (1a, b, c pg. 114) and work as a whole group together to create these. Then for 1d, have participants as groups build a poster for that strategy. The posters should have three components: a verbal description of the strategy, a representation of the strategy (using base ten models or number lines) and a story context. Assign each component to some portion of the group. Provide 5 minutes for participants to work, then solicit their responses and create a poster as a group.
When the poster is complete ask the group what mathematical ideas are evident in this work.
Remember that verbal descriptions are not number specific and students will need strong math vocabulary. (The verbal description is important, because it needs to be process-specific and not number-specific. This is what we have to teach students to do as part of our classroom work. It takes a lot of math vocabulary)

97. Small-Group: Addition and Subtraction Strategies
Creating Subtraction Posters
15 minutes (12:55-1:10) (page 107 facilitator's guide small group)
Distribute the Math Activity sheet (pg. 114).Section 2
Have each group focus on two of the five given subtraction strategies. Everyone does 2c and one other one (presenters will assign). The same process we did whole group should be done for each strategy assigned.
As you interact with the small groups, take note of approaches that you want shared in the whole group. After 20 minutes announce that each group should be working on their posters for the strategies assigned to them.

98. Gallery Walk: addition and subtraction strategies
10 mins (1:10-1:20)
Teachers walk around and look at the posters that the groups created. Be sure to place posters for the same strategy together. At the end of this time, allow teachers to share any big "aha's" that they had. What is the same or different between the strategies? What mathematical ideas are present in this work?

99. Mathematical Goals for Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
Mathematical Goals for
Sessions Three
5 minutes (1:20-1:35)

100. Break!
15 mins (1:25-1:40)

101. DVD: Addition and Subtraction
25 minutes (1:40-2:05) ( page 108 Facilitator's Guide)
The DVD provides glimpses into many different classrooms grades K-4 and offers images of students working on addition and subtraction problems. It also includes interviews with teachers about how their students develop a system of tens. Let the group know that you will view the entire DVD at one time. Ask participants to make notes of any points they want to discuss and include those in a brief discussion after the viewing of the DVD.

102. Mathematical Goals for Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
5 minutes (2:05-2:10)

103. Small Group Discussion: Addition and Subtraction Algorithms
Case 14
Focus Questions 3 and 4
30 minutes (2:10-2:40) (Page 109 Small Group Facilitator's Guide)
Distribute Focus Questions: Chapter 3, p. 115 Case 14 Nadine (page casebook)
As participants discuss these cases, they should sort out the thinking of the students' approaches and determine what mathematics these procedures are based on before discussing the pedagogical issues that these teachers raise. These cases should support participants as they solidify their own mathematical ideas about addition and subtraction of multi-digit numbers and should build on the previous math discussions.
The other 2 cases can be brought to the job embedded table as it is deemed necessary by the presenters.

104. Whole Group Discussion: Addition and Subtraction Algorithms
• What is the same about the two strategies in case 14?
• What is different about the two strategies?
• What are the mathematical
principles underlying each of
the strategies the students
use?
15 mins of the 30 minutes (2:40-2:55) Facilitator's Guide Page
Use the guiding questions in the slide for the whole group discussion.

105. Mathematical Goals for Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
5 minutes

106. Task #1 - Read and discuss the article "Orchestrating Discussions"
Task #2 - Do the Writing Assignment: A math interview
One assignment should be done each quarter.
Article -Orchestrating Discussions - Task 1
Writing Assignment Math Interview - Task 2
Distribute the Homework, p. 75 facilitator's guide
This work will be followed up on in JEPD.

107. Exit Cards.............. • What was important or significant to you in
the mathematics discussed at this session?
• What mathematics are you still wondering about from this session?
• What do you want to tell us about
how the seminar is working for you?
(See Facilitator Guide page 110)