1. Reflecting on Practice: Mathematics and Motivation
Session 3
PCMI Outreach
Reflecting on Practice
Park City Mathematics Institute
Mathematics and Motivation
Session #3 Focus: Accountability Routines
Goals for the day: Think about how to create the accountability norms of everyone participates, listening matters, and the focus is on mathematical ideas by using debate structures, number talks, and smudged math.
Participants will
Unpack what accountability means in the context of motivation
Experience three routines that highlight classroom accountability
Explore how the three routines connect to accountability and how the routines might fit in teachers’ classrooms
Materials needed: PPT Two Day RoP Session 3 Accountability 2018 PCMI CYCoM
Vertical non-permanent surfaces & dry-erase markers, speakers for video
Handout: Student Work for Debate
Video:
Handout: Smudged Math Problems
References: Encouraging Debate. PBS Learning Media.
Horn, L. (2017). Motivated. Portsmouth RI: Heinemann
Luzniak, C. (2015). Debate Resources.
Smudged Math
Materials photocopied: Handout: Student Work for Debate
Handout: Smudged Math Problems

2. Park City Mathematics Institute
Welcome Back!
Take a few minutes to introduce yourselves at your table- not only your name and where and what you teach but also-
What’s your favorite math movie? OR
What are you doing while you’re grading papers?
Reflecting on Practice
Park City Mathematics Institute
0:00 5 min (slide 2)
Community Building Activity (start of Day 2)
Take a few minutes to introduce yourselves at your table- not only your name and where and what you teach but also-
What’s your favorite math movie? OR What are you doing while you’re grading papers?

3. RoP: Student Motivation
Teachers will leave with a framework for thinking about motivation & strategies to help students want to engage with mathematics.
Specifically, we will focus on:
Meaningfulness
Belongingness
Accountability
Reflecting on Practice
Park City Mathematics Institute
0:05 5 min (slides 3-6)
The first session we looked at meaningfulness. The second we looked at belongingness. For our final session, we will look at accountability.
We are defining accountability not as an assessment system or teacher accountability for test scores, but as student accountability to learning. We want students to feel like they are part of a community where learning is meaningful, where they can be competent in multiple ways, and where they feel a sense of accountability to participating in learning because of their connection with their teacher and their peers.

4.      
According to Lani, not only are there five motivational features in what she calls a Motivated classroom, but there are many interconnected features of motivation. While we are focusing on meaningfulness, belongingness, and accountability separately, each perspective builds off of the others, and many of the strategies we are talking about address multiple areas of her framework.

5. Park City Mathematics Institute
Accountability
“Accountability refers to the structures and routines that oblige students to report, explain, or justify their activities. Often reduced simply to assessment, accountability goes beyond how we grade to encompass the routines and norms that enjoin students to participate in particular ways in classroom life. When students feel a sense of investment in and accountability to their classmates, for example, this changes the risk-benefit calculus, leveraging positive peer pressure to increase participation.”  (Horn, p. 8)
Reflecting on Practice
Park City Mathematics Institute
Show the slide with a definition of accountability from Lani.  
This is Lani’s definition of accountability.

6. Norms of Participation
are about the expectations students and teachers have for the way class works: 
1) everyone participates,
2) listening matters, and,
3) the focus is on mathematical ideas.  
Reflecting on Practice
Park City Mathematics Institute
Lani emphasizes three norms in the ways students participate that create a culture of accountability. These three norms are above; the routines we’ll look at today are meant to create and reinforce these norms.

7. A Debate The question is: “Whose solution is the best and why?”
To prepare for your table's debate, spend a few minutes making observations about something you like about each solution
e.g. “I like that they made a table”. 
Reflecting on Practice
Park City Mathematics Institute
00: min (slides 7-12)
Note that Lani talks about accountability as the “structures and routines that oblige students to report, explain or justify their activities.” 
We are going to engage in three different structures/ instructional routines and think about how they might look in our own classrooms.
The first routine is a debate: The idea is that kids would be given student work and have this debate.
For our first routine, we will be putting our student hats on for the task. Spend 5 minutes interpreting the question, “Whose solution is the best,” as you think students might approach it.
Pass out debate handout: Student Work for Debate
Anticipated Responses:
What was good:
Student 1: knew that 24 was 16; knew to split 20 into 16 and 4
Student 2: knew to split 2x+2 into (2x)(22); knew to distribute common factor 2x
Student 3: knew what was supposed to happen with the numbers; justified reasoning
Student 4: knew graphing could help find solution; knew how to use the technology
Student 5: knew how to solve the simple linear equation
Which one is best
Some might argue for the graph made by student 4 because it is a quick way to find the solution and shows understanding of the power of graphical representations but others might argue that do not know just by looking at that window there is only one solution
Some might argue for solution made by student 2 because it is the expected approach and correctly applies an understanding of exponents but others might argue student might just be following a procedure without and understanding
Some might argue for the table made by student 3 because clearly knows what a solution should do and is methodically working toward finding one and reasoned why there are no other solutions; others might argue that the reasoning did not include fractional exponents- not sure what would happen with x= 2 ½ for example.

8. and my WARRANT is ____________,
Debate Structure
At your tables, debate using the following structure:
Make a CLAIM and a WARRANT to support your claim
           
My CLAIM is __________,
  and my WARRANT is ____________,
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At tables, have this discussion using this structure of Claim/Warrant
Facilitators should watch the video ahead of time to make sure they know what the debate structure looks like.
Chris Luzniak has between 3-5 people share their arguments.

9. Park City Mathematics Institute
Chris Luzniak (PCMI '10 & '11)
Reflecting on Practice
Park City Mathematics Institute
We are going to watch a class engaged in this activity
This is the Unit Circle they are considering.

10. Park City Mathematics Institute
Chris Luzniak (PCMI '10 & '11)
PBS Video 
b/encouraging-debate#.W0aB69JKhPY
Reflecting on Practice
Park City Mathematics Institute
Play from "0:50 to 4:00”

11. These are some recommendations for writing debateable questions, from Chris Luzniak, the teacher in the video.
If teachers think these prompts are too vague to have a debate; Please remind participants that the goal of debating in this case isn’t necessarily to win an argument, but to listen and explore perspectives.

12. Park City Mathematics Institute
Take a few minutes with your table group and talk about how this routine could encourage participation in your classroom.
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Small group discussion
Make reference of Chris Luzniak's debate structure in the book.  

13. Park City Mathematics Institute
Solve the system
x + y = 10, y = x + 2
Reflecting on Practice
Park City Mathematics Institute
00: min (slides 13-16)
Number Talk
Here is a second instructional routine or structure: 
Have any of you ever taken part in a number talk? Great. We are going to try the same approach in an “algebra talk.”  This could also be called an “algebra string.”
x + y = 10, y = x + 2
y + x = 20, y = x + 2,
x + y = 20.5, y = x + 2,
Facilitators write the first set of equations up on the board. Participants use their hands to indicate if they have one solution strategy (one finger), two strategies (two fingers), or more (more fingers).  When a majority of the room has their hands showing at least one strategy, the facilitator will call on a participant. The facilitator will record  the strategy on the board.  After hearing the first solution, the facilitator will prompt the group for another strategy, and another if available, and record those strategies. Then, the facilitator will write the second set of equations up on the board and ask participants to repeat the process. Again, participants will share solutions, and then proceed to the third string.   
The facilitator should label each strategy with the participant’s name, and leave all the strategies up for people to draw on as they solve the successive problems.
Anticipated Responses:
I used substitution, and made it 2x+2=10, solved to get 4
I tried to figure out which two integers were two apart that added to 10
I knew they needed to be two apart, so I cut 10 in half, added 1 and subtracted 1, and got my solution
I subtracted x on each side, used elimination to get 2y=12, then y = 6

14. Park City Mathematics Institute
Solve the system
x + y = 10,  y = x + 2
x+ y = 20,  y = x + 2
Reflecting on Practice
Park City Mathematics Institute

15. Park City Mathematics Institute
Solve the system
x + y = 10,  y = x + 2
x + y = 20,  y = x + 2
x + y = 20.5,  y = x + 2
Reflecting on Practice
Park City Mathematics Institute

16. Park City Mathematics Institute
    Take a few minutes with your table group and talk about how this routine encourages participation in your classroom.
Reflecting on Practice
Park City Mathematics Institute
After going through the three examples, have participants individually write down several advantages or disadvantages of having number talks as part of their classroom routines. Then, allow time for teachers to share out how they might use this. Note that many teachers may already use number talks, and may have strong opinions about best practices or strategies that work for their students.
Anticipated Responses
Advantages:
Opportunity to encourage/reinforce wait time
Develops numeracy strategies
Students can make connections between multiple approaches to a problem
Student participation encouraged through thumbs up indication that they have the right answer
Students have to communicate 
Disadvantages:
Students can “get by” without volunteering an answer or being called on
Students might feel “at risk”; they have to come up with something
Some students might never get the idea
Hard to do a talk around some topics
Some people need to write things down to think about them

17. Smudged Math Can this equation be true? 00:45 20 min (slides 17-22)
Smudged Math (#smudgedmath) is something shared by Peter Liljedahl and many other folks as a way to modify tasks to help lower the floor so more students can participate and create opportunities for discussion or creating arguments. The idea is broad – smudging some type of equation, diagram, or mathematical statement in a way that creates ambiguity, gets at a mathematical idea, and can start conversation.
Notes for facilitators:
Participants might ask: do the smudges have to be the same? Can I find all of the solutions? Can I prove I found all the solutions? Any of these questions could lead to useful thinking, and the task is intentionally framed ambiguously so that students can enter at a low floor – for instance by arguing that it is or isn't possible without finding examples, and beyond that finding one or several examples, or trying to find all the examples.
Peter Liljedahl would likely frame this as a question meant to get students thinking, so if teachers say, "well my students might just say "it's possible" and stop there, it could be worth talking about how part of the Thinking Classroom framework is having hints and extensions ready to prompt students to do more thinking with the task. For this task, that might look like prompting students to find more solutions, and then have them try to prove that they found all solutions. Another extension is exploring if both smudges are the same.
Reflecting on Practice
Park City Mathematics Institute

18. Norms of Participation
are about the expectations students and teachers have for the way class works: 
1) everyone participates,
2) listening matters, and,
3) the focus is on mathematical ideas.  
Reflecting on Practice
Park City Mathematics Institute
The purpose of this structure is to reinforce these norms. The structure helps to lower the floor, so that more students can participate in different ways. Students must make sense of the problem and compare different approaches, which reinforces the norm that listening matters. Finally, these problems don’t have easy procedural solutions, encouraging a focus on mathematical ideas.

19. Park City Mathematics Institute
On your own
Choose one or two of the problems on the handout.
Solve it/them.
If you have time remaining, think about an extension or how you might adapt these for your classroom
Handout: Smudged Math Problems
Using the ten smudged problems, participants will choose one or two questions that they will work and solve individually.
When finished solving, spend a few minutes thinking about an extension or how you might adapt these for your classroom
Reflecting on Practice
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20. Park City Mathematics Institute
On the board
Head to VNPSs!
Share with your card-mates your solutions and/or extensions.
Be sure to be actively listening, as you'll be bringing this discussion back to your table.
You'll be sharing one of the solutions/extensions you heard with a table partner in a few minutes
Emphasize that discussions at boards should focus on what they liked about the problems they looked at and how they might use these problems or similar problems in their classrooms. Participants will be held accountable to sharing something useful they heard when they return to their tables.
Reflecting on Practice
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21. Park City Mathematics Institute
Back to the tables
With a partner, back at your table, share at least one interesting thing someone else at your board presented
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Park City Mathematics Institute

22. Smudged Math Debrief
How does having students use this routine of smudged math and/or active listening relate to accountability?
Norms of Participation: 
- everyone participates, 
- everyone has to listen to how others thought about the problem, and 
- the focus is on important mathematical  ideas.  
Debrief Activity
At your tables, take a few minutes and consider the question on the slide. Then use a round robin protocol for the discussion.
How does giving students this kind of peer feedback relate to accountability?
If discussion runs dry, perhaps add the following
-What would this look like in your classroom?
-What would be the advantages/disadvantages
Anticipated Responses: 
Everyone is responsible for reading and responding to someone else’s work
Opportunities to learn from the ways their partner worked on a problem
Students have to communicate what they were thinking in their critique
The questions to use in reading the work of others establishes some norms for discussing work- “This confused me, could you give more detail, not sure I understand the terms”
Select participants to share out for a short discussion
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23. Park City Mathematics Institute
Reflection Question
What are the gaps between what you say you value and what students do?  
What structures or routines can you incorporate that might help you teach your students new ways of being in math class?
01: min
What structures or routines can you incorporate that might help you teach your students new ways of being (accountable) in math class?
Reflect on your own classroom and journal on this question for a few minutes. After a few minutes, share some thoughts with your table.
[Adjust this as needed based on time]
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24. Park City Mathematics Institute
One Goal
In respect to what we have learned today, write down one goal that you have as an enhancement to your teaching practice.
What are some ways that you can begin to implement your goal?
Share your ideas with a partner or at your table.
Reflecting on Practice
Park City Mathematics Institute
01:10 5 min
In respect to what we have discussed and learned, write down one goal that you have as an enhancement to your teaching practice.
What are some ways that you can begin to implement your goal?
Share your ideas with a partner or with your table.
[Participants can use the index cards to write down their goal and have for a reminder note in their classroom.]

25. Thanks for a great weekend!

26. Park City Mathematics Institute
References
Encouraging Debate. PBS Learning Media.
Horn, L. (2017). Motivated. Portsmouth RI: Heinemann
Luzniak, C. (2015). Debate Resources.
Smudged Math
Reflecting on Practice
Park City Mathematics Institute