1. Coaching for Math GAINS Peel Co-Teaching Project Anchor Session #2
April 1, 2010 1

2. Reviewing our Norms Start and end on time.
Contribute to a safe learning environment that encourages risk taking; be kind.
Listen actively; speak fearlessly.
Invest in your own learning and the learning of others.
All electronic communication devices off except during lunch/break.
Suffering is optional!

3. Overview of the Morning
Welcome
Responding to Feedback
How well have you been listening?
Sharing your journey
Mathematics as a social activity
Is your locker open?
Common Questions … consolidating questions for parallel tasks

4. We were listening … Working on open & parallel tasks
Time to dialogue and plan with colleagues
Logistics … how do we make it work?
Opening up my classroom to others
Purpose of Math GAINS project
Ministry perspective
Peel perspective
What the research says
Making sense of the "Big Ideas"
Based on the feedback that we received from you on the Cheers, Fears & Unclears exit cards …
You said that you really liked … so we've built in opportunities during today's session to …
You said that you had concerns about the logistics of making the project work; about teaching in front of colleagues; the role of the provincial facilitators … it appears that most of those concerns have been addressed within your families of schools teams, and that every family has developed a working rhythm that works for them … opportunity to share additional concerns today?
You said that you were unclear about the purpose/goals of this project, and how to make sense of the big ideas, so …

5. Peel Perspective - Math GAINS Project
Facilitated co-teaching activities to
promote job-embedded professional learning opportunities related to supporting the implementation of the ongoing priorities and initiatives of the
Peel District School Board
These include,
but are not limited to the following –

6. Peel Perspective - Math GAINS Project
Transformational Practices
School Success Planning
Differentiated Instruction
Cross-Panel Connections
Student Success / Learning to 18
Transitions (Grade, Panel, Course)
Report Card for Student Success
Alternative Programs

7. Research Foundation for PD
Context
Providing sufficient time;
Engaging external expertise;
Engaging teachers in the learning process;
Challenging discourses;
Providing opportunities to interact in a community of professionals.
Math GAINS
Eight release days;
Math Coach;
Teachers working together in school teams and working cross-panel;
Rethinking our approach to teaching math;
Co-planning days, anchor sessions and debriefing after demonstration lessons.

8. Research Foundation for PD
Content
Different aspects of content are integrated;
Clear links made between teaching and learning and student teacher relationships;
Assessment is used as a focus;
Sustainability of improved student outcomes.
Math GAINS
Big ideas, questioning, group work skills and problem solving;
How we teach math impacts on how well students learn and we can connect through more engaging tasks;
Formative assessment using open and parallel tasks;
Students can continue to improve.

9. Research Foundation for PD
Activities
Content and activities aligned;
A variety of activities needed;
Professional instruction sequenced;
Understandings discussed and negotiated.
Math GAINS
Developing then trying open or parallel tasks;
Different tasks, grades and strands covered;
Modelling, co-planning then teaching
Active discussion with colleagues of what works for students.

10. Research Foundation for PD
Learning Processes
Substantive change is difficult;
Some new understandings are consistent with current thinking and some are inconsistent;
Teachers can learn to regulate their own learning.
Math GAINS
Trying questions together;
Some tasks are similar to what we have done before, some are a new approach, with support see what works for students
Begin to use questioning in other lessons if it supports student understanding.

11. How well have you been listening?
We spend a great deal of time talking about questioning.
How effective some of our questions will be, really depends on how well we listen to our students.
The more we listen … the more our role will evolve from "Stand & Deliver" to …
We've been talking a great deal about questioning … open & parallel questions to pose in the Minds On or Action phases of the lesson; common questions to ask after students have worked on those types of questions; powerful questions to get at a student's understanding of the lesson goals or the big ideas; probing questions to evoke & expose a student's thinking; prompting questions to help them move forward, without removing their need to think.
As we continue to ask better questions, and create richer learning experiences for our students … we will gradually move away from our traditional (expected) mode of delivering curriculum content … Stand & Deliver.

12. Impress Me!
As you watch this video, please consider the following questions:
How does the participating teacher describe her usual classroom practice?
What is new/different for the teacher and her students in this lesson?
How do the students respond to the lesson approach?
4. What observations/reflections does the teacher share in the lesson debriefing?
5. How does your co-teaching experience compare with this example?
Caroline Price is a ninth grade teacher at the Ravensbourne School in West Bromley School District in London, England. She share's her experience of abandoning her traditional "Stand and Deliver" approach.
After watching the video:
Even if your classroom practice already follows many of the strategies shown in this video, you can still ask yourself these questions (so that teachers who already engage students in co-operative learning realize they too can still add more to their practice).
Only the teacher can answer the first question, which is why the hosting teacher drives the co-teaching cycle.
The observing teachers can help with the second question, so that the hosting teacher can focus mostly on the lesson delivery during this process.

13. Self Reflection “What can I add to my practice to help
A big idea of co-teaching is to reflect on your current classroom practice, whatever that may be, and ask:
“What can I add to my practice to help
students understand the concepts I
am teaching more deeply?”
“How will I know if my changes are
having a positive impact on my students?”

14. Sharing Your Experience
Each person in your team will go to a different location.
With the other people at your location, form an inside/outside circle.
Each "inside" person will start. Choose one question to ask from the following list. Take turns.

15. Sharing Your Experience
What was the focus of the lesson that was co-planned? How was the lesson delivered? What was the most valuable feedback from the debrief?
How did your students react to the lesson? How did they feel about all of the observers present?
The biggest surprise of co-teaching is …
A change in your practice that has resulted from your involvement in the GAINS project is …
Starting with the person from the inside circle, ask your partner one of these questions. Partner will have 2-3 minutes to respond. Then the people on the inside will rotate clockwise two people.
Inside people ask questions during the first three rotations … outside people will ask during the final two rotations.

16. Math as a Social Activity
As you watch this video clip, record the steps the teacher takes to create the learning environment he wants for his students.
We talked earlier about how important it is for us to become better listeners, if we want to ask better questions. It is also important, that our students operate in an environment where they feel confident to verbalize what they are thinking to their peers and to you, the teacher. The following video clip outlines one teacher's attempt to create a positive learning environment for his students. 16

17. Break Time!
Approximate break time: 10:30 – 10:45

18. The Locker Problem
There are 1000 lockers in the long hall of the Peel District High School. In preparation for the beginning of school, the janitor cleans the lockers and paints fresh numbers on the locker doors. The lockers are numbered from 1 to 1000.

19. The Locker Problem … continued
In your family of schools team, identify 3 people who will act as observers.
The remaining team members will work on the problem in "the fishbowl". They may divide themselves up into smaller groups if they wish to.
Send one of the observers to the materials table to collect the forms they will use to record their observations.

21. The Locker Problem … continued
Explore the problem in your groups. Various manipulatives are available for use from the materials table.
Record your findings on chart paper.
Be sure to explain the mathematics that will justify why your answer is correct.

22. The Locker Problem … continued
When the school's 1000 students return from summer vacation, they decide to celebrate the beginning of the school year by working off some energy.
Student #1: opens every locker.
Student #2: starts at locker #2 and closes every 2nd locker.
Student #3: starts at locker #3 and opens or closes every 3rd locker.
This process continues ... until all 1000 students have entered the school.
Which locker doors are open once every student has arrived?

23. Consolidation of the Locker Problem
Life in the Fishbowl  observers share their observations with their team members
Presentation of Solutions  a math congress 23

24. Parallel Tasks Revisited
Focusing Our Lens on
Common Questions

25. Possible Lesson Goals Find the distance from a point to a line segment
AND / OR
Apply the geometric properties of circles, midpoints, line segments, and perpendicular lines to the real world

26. Consider this example:
The diagram shows the locations of Katie’s, Krista’s and A.J.’s homes.

27. The Parallel Tasks Option 1 Option 2
A. J. and Krista want to meet somewhere that is equally distant from each of their homes. Where could they meet? OR
Option 2
Katie, Krista, and A. J. want to meet
somewhere that is equally distant from all three of their homes. Where could they meet?

28. Possible Tools GSP Grid paper Rulers
Soft measuring tapes and/or string

29. The Solutions Option 1 There are actually many solutions to Option 1.
All the points on the perpendicular that meets the middle of the line segment between A.J.’s and Krista’s house are equally distant from each house.

30. The Solutions
Option 1

31. The Solutions Option 2 There is only one solution to Option 2.
Students need to find the single point at the centre of the circle that passes through all three of the points representing the locations of the homes.

32. The Solutions
Option 2

33. Meeting Student Needs
The first option allows students the opportunity to apply their knowledge of lines to solve a real life problem.
The second option allows those students needing a challenge the opportunity to apply their knowledge of lines and circles to solve a real life problem.

34. Other Variations
Choose points with positive coordinates for students who struggle with integer calculations, and points with negative coordinates for those who do not.
Choose points with whole number coordinates for students who struggle with fractions or decimals, and fractional coordinates for students who do not.

35. Common Questions
Whichever task the students complete, the teacher could ask:
How did you find the meeting spot? Is there another way to solve the problem?
What tools did you use to help you?
Is there more than one possible meeting spot? How do you know?
How could you verify your meeting spot is equally distant from all three homes?

36. Principles to Keep in Mind
Parallel tasks need to be created with variations that allow struggling students to be successful and proficient students to be challenged (consider common student stumbling blocks when creating your tasks).
The tasks and common questions should be constructed in such a way that will allow all students to participate together in follow-up discussions.

37. Consolidating Questions
Consolidating questions can be used to tie big ideas together at the end of a lesson or activity.
Common questions can serve as consolidating questions when the main activity of a lesson has been a parallel task.

38. Your Turn! There are six different parallel tasks at your table.
Work in groups of 2 or 3.
Choose a task.
Solve the problem.
List student stumbling blocks that might be addressed by offering each choice.
List common questions that could be asked of all students at the end of the task.
Record your work on chart paper, and post according to tasks.

40. Welcome Back!
Effective teaching involves risk taking … by both the teacher and the student.

41. Overview of the Afternoon
Connecting Big Ideas with Expectations and Lesson Goals
Consolidating Questions
Time to Practice & Share
4. A Few Words About Logistics
5. Group Communication: SharePoint

42. Why use big ideas?
By thinking about the big ideas, it becomes easier to develop appropriate lesson goals and appropriate consolidating questions to bring them out. 42 42

43. Comparing patterns helps us see there are classes of patterns… (BI5)
For example …
If you are thinking about the Big Idea:
Comparing patterns helps us see there are classes of patterns… (BI5) 43 43

44. … and your goal is to have students understand what linear growing patterns are, you are likely to ask:
Which two of these patterns do you think are most alike and why:
5, 8, 11, 14, 17, 20,…
5, 10, 20, 40, 80,..
5, 10, 15, 20, 25,..
5, 20, 35, 50, …? 44 44

45. … rather than simply asking for a definition of a linear growing pattern.
Why … because you want them to be able to identify the distinguishing characteristics of a linear growing pattern. 45 45

46. Building lesson goals
You can use a big idea to hone in on an appropriate lesson goal. 46 46

47. Relationship among Expectations, Big Ideas, & Lesson Goals
Topic
Big Ideas
Might start with topic– put through sieve of big ideas, get a goal and check against expectations; might start with expectations instead
Expectations
Goals 47 47

48. Relationship among Expectations, Big Ideas, & Lesson Goals
Sometimes you can reframe the big ideas for your topic. For example, a trig big idea might be: 48 48

49. Relationship among Expectations, Big Ideas, & Lesson Goals
Limited information about a periodic relationship can sometimes, but not always, reveal other information about that relationship. 49 49

50. Or…
When a relationship appears to be periodic in nature, then it is appropriate to consider a trigonometric function to model the relationship. 50 50

51. Relationship among Expectations, Big Ideas, and Lesson Goals
We will use the Posing Powerful Questions Template (PPQT) as a tool. 51 51

52. An Example … Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Determine other representations of a linear relation, given one representation.
Big Idea(s) Addressed by the Expectations
Goal(s) for a Specific Lesson 52

53. What Big Idea is being Addressed?
Most likely BI 4 53 53

54. List the Big Idea(s) … Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Determine other representations of a linear relation, given one representation.
Big Idea(s) Addressed by the Expectations
Different representations of relationships or patterns show different things about them and which is more useful depends on the situation.
Goal(s) for a Specific Lesson 54

55. Create Your Lesson Goal
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Determine other representations of a linear relation, given one representation.
Big Idea(s) Addressed by the Expectations
Different representations of relationships or patterns show different things about them and which is more useful depends on the situation.
Goal(s) for a Specific Lesson
Students will recognize when a graphical model is more useful and when an algebraic one is more useful. 55

56. What does this mean for consolidating the lesson?
You need to ask a question or two that gets RIGHT to your goal. 56 56

57. A Consolidation Question Might be:
Lesson Title:
Grade/Program: 7
Consolidate/Debrief Sample Question(s)
You have a graph with x-values from -10 to 10 plotted. You want to know the values of y for specific values of x. For which values of x would you use the graphical form? For which values of x would you use the algebraic form?
Curriculum Expectations
Determine other representations of a linear relation, given one representation.
Big Idea(s) Addressed by the Expectations
Different representations of relationships or patterns show different things about them and which is more useful depends on the situation.
Goal(s) for a Specific Lesson
Students will recognize when a graphical model is more useful and when an algebraic one is. 57

58. Consider the expectation:
Another Example
Consider the expectation:
“Solve first degree equations with non-fractional coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1)…” 58 58

59. Which big idea do you think it most closely relates to?
Another Example…
Which big idea do you think it most closely relates to? 59 59

60. Another Example… BI 4 You might have picked
thinking that solving an equation means representing it in a different, easier to recognize form. 60 60

61. Another Example… BI 6 You might have picked
thinking that you had some information that could give you other information. 61 61

62. Another Example… BI 3 You might have picked
thinking that an equation is a way to describe a change and solving it is just about “undoing” the change. 62 62

63. What might the lesson goal be?
Your lesson goal should be informed by which of those big ideas you want to focus on. 63 63

64. Option #1 Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Different representations of relationships or patterns show different things about them and which is more useful depends on the situation.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means determining an equivalent equation where the solution is more obvious. 64

65. To clarify what this means …
These equations are equivalent:
x = 4
2x – 7 = 1
3x + 7 = x + 15
But, it's sure easier to see the unknown in one of them! 65 65

66. Option #1 Consolidate/Debrief Sample Question(s)
Lesson Title:
Grade/Program: 7
Consolidate/Debrief Sample Question(s)
Agree or disagree: The equation 5x – 4 = x is really the equation x = 10.5 in disguise, just easier to solve. OR Why might someone say that solving an equation is about finding what easier equation is being disguised?
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Different representations of relationships or patterns show different things about them and which is more useful depends on the situation.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means determining an equivalent equation where the solution is more obvious. 66

67. Option #2 Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Sometimes knowing a few things about a pattern or relationship allows you to predict other things about that pattern or relationship.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means that you know some information (an output and a rule), so you should be able to figure out the other information (the input). 67

68. Option #2 Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Sometimes knowing a few things about a pattern or relationship allows you to predict other things about that pattern or relationship.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means that you know some information (an output and a rule), so you should be able to figure out the other information (the input).
Consolidate/Debrief Sample Question(s)
You know one of these two things: x + 2y = 20 OR 3x + 2 = 20
Which one lets you figure out what the value of x is? Why? 68

69. Option #3 Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Algebraic reasoning is a process of describing and analyzing
generalized mathematical relationships and change using words
and symbols.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means using the change rule embedded in the equation, symbolically, to describe one specific example of the effect of the change. 69

70. Option #3 Curriculum Expectations
Lesson Title:
Grade/Program: 7
Curriculum Expectations
Solve first degree equations with non-fraction coefficients using a variety of tools (e.g. 2x + 7 = 6x – 1).
Big Idea(s) Addressed by the Expectations
Algebraic reasoning is a process of describing and analyzing
generalized mathematical relationships and change using words
and symbols.
Goal(s) for a Specific Lesson
Students will recognize that solving an equation means using the change rule embedded in the equation, symbolically, to describe one specific example of the effect of the change.
Consolidate/Debrief Sample Question(s)
A change rule suggests that you triple a number and add then 2 to it. What equation would you solve to find the input, if you know the output is 23? How would you solve it? 70

71. What's Really Important is …
getting a goal clear in your own mind. This can make a big difference in increasing the likelihood that students will learn what you hope they will learn. 71 71

72. That includes knowing why you have that goal.
Use an example, e.g. your goal might be that students can use order of operations, but why is that your goal?
If you are clear about your goal, and you know why you have that goal … then you are more likely to know what questions to ask, what activity to use,… 72 72

73. Make it yours
Even if you get a lesson from a valued resource, you have to make your OWN decision about what to pull out of that lesson. 73 73

74. Now… Choose a lesson in a resource that you brought. OR
Choose one of the PPQT lesson outlines on your table. 74 74

75. Let’s consolidate Find a partner that you have not worked with before.
Work with your partner to restate the goal to focus more clearly on a big idea.
Write a consolidating question to match your goal.
Share what you've done with the other members of your team. 75 75

76. Here's Your Challenge With a partner, choose 1 or 2 of these goals.
Focus them to relate more explicitly to one or more of the big ideas.
Write consolidating questions to match your goal.
Use the PPQT to record all of your thinking.
We talked earlier about how consolidating questions should tie to lesson goals. They should tell you if your goal was achieved. 76 76

77. Consider These Lesson Goals
Represent a relation using a table of values, a graph or an equation.
Identify direct and partial variations.
Identify properties of linear relations.
Represent a linear relation in a different form.
Recognize whether a relation is linear or nonlinear.
The preceding stated goals were taken from a series of lessons on linear relations in a grade 9 text. 77 77

78. Logistics - Math GAINS Project
PAM Code 59 is to be used to book supply coverage for any release time related to this project
when using PAM Code 59, please provide Krystal Wilson and Alan Jones with relevant details and information (who, when, where, what)
<names of teachers> from <name of school> will be attending a co-planning session at <name of host school> on <date of co-planning session>

79. Logistics - Math GAINS Project
up to 10% of our overall funding allocation may be used to purchase resources to support the co-teaching activities
approximately $1000 is available for each school involved in the project to purchase instructional and professional resources
possible resources include thinking tools such as manipulatives and technology, and/or professional resources to support questioning, big ideas, differentiated instruction, etc.
completed purchase orders are to be submitted to Krystal Wilson and Alan Jones

80. SharePoint

81. Don’t forget your homework!
Thank you.
Don’t forget your homework!