1. Module 2: Prompting Argumentation
Focus on Tasks
June 28, 2016
“Children must be taught HOW to think not WHAT to think.”
-Margaret Mead
6/2816

2. Do Now: Triangle Problem
DO NOT SOLVE: Discuss how to solve the following problem
Belinda leads
>> idea claim not obvious
>> assumption – is it any triangle, only equilateral
Does your argument show it’s true for ALL triangles
>> facilitated at the tables

3. Community Agreements Judgment Free Zone Stay engaged
Stay focused. Be present and engaged
Be on time and stay on task
Respect others’ opinions/thoughts
Leave your cell phones alone (be present)
Listen Attentively
Stay Positive
Stay engaged
“Vegas” Rule
Encourage risk taking
Be positive
Be respectful of each other
Be positive and encouraging
The “Photo Album” rule

4. Objectives
Teachers Will Be Able To:
Develop a deeper understanding of argumentation and its potential in the math classroom.
Select & Modify tasks to prompt argumentation for a variety of instructional purposes by using three conceptual lenses.
Evaluate & Analyze tasks to determine how they support argumentation in the math classroom.
So that Teachers Can:
Implement tasks that prompt argumentation to support a culture of thinking.

5. Brainstorming Answer individually…
Then discuss and answer as a small group…
Then, we will share as a whole group.
What is the value of argumentation in the classroom?

6. What is the value of argumentation in the classroom?
Brainstorming
What is the value of argumentation in the classroom?
Active engagement.
Students controlling the math thinking.
Informative for teacher grouping students
Supports student respect and confidence in work – worth considering more.
Deepening understanding
Social interaction promotes lifelong skills: reasoning, logic, communication, value multiple points of view
Culture of thinking supports comfort sharing. Less worry about being wrong.
Mathematical literacy – terminology linked with sense-making.
Builds future learning – broader practice of constructing an argument
Helps teacher to be better observer
Requires different planning – fewer problems, more time.
Slower speed supports range of learners
Promotes stronger, more personal connection to content.
Perseverance in problem solving
Students who might be less proficient can help those who move more quickly to think deeply and connect to underlying steps and concepts
Diagnostics of underlying issues – car repair analogy

7. Do our Math Tasks need a Makeover?
High school teacher Dan Meyer shares thoughts about math tasks often shared with students in the U.S.
First 6-8 mins of video

8. Lenses for thinking about tasks
Lens 1: (Engagement)
Does the task engage students in mathematical argumentation? How can it be modified?
Lens 2: (Student Purpose)
What do you want the students to learn from the mathematical argumentation task?
Lens 3: (Teacher Purpose)
What will you learn about students by using the mathematical argumentation task?

9. Lens 1: What are the look fors?
Lens 1: (Engagement)
1. Does the task promote mathematical discourse? A conversation?
2. Does the task prompt students to articulate a chain of reasoning?
3. Does the task require students to write or otherwise record their chain or reasoning to show the result is true?

10. Is it a 1/4? Problem: Lens 1
Lens 1: Does the task engage students in mathematical argumentation?
What fraction of the rectangle below is shaded?
Original
Revision
Laura says that ¼ of the rectangle is shaded. Do you think she is correct? Write an argument to support your answer.

11. DJ Problem: Lens 1
Lens 1: Does the task engage students in mathematical argumentation?
Original
In preparation for the Prom, students are researching the costs of two local DJ companies. Music Makers charges a fee of $200 and an additional $175 per hour. Dance Partners does not charge an initial fee, but charges $225 per hour. Which company would be more cost effective for the prom committee?
Revision
In preparation for the Prom, students are researching the costs of two local DJ companies. Music Makers charges a fee of $200 and an additional $175 per hour. Dance Partners does not charge an initial fee, but charges $225 per hour. Which company would be more cost effective for the prom committee?
Write a mathematical argument to support your decision.

12. Your Turn… Break Out Groups: Elementary & Secondary
1. Examine the example tasks on the handout
2. Use Lens 1 to think through each task
Does the task engage students in mathematical argumentation?
3. Generate possible revisions for the task

13. Your Turn, Lens 1: Engaging Mathematical Argumentation
Lens 1: Does the task engage students in mathematical argumentation?
Solve each of the following (on a worksheet)
a) = ____ b) 500 ÷ m = c) 25 x 10 x 4 = _____
a) 3x+5 = 2x-6 b) 4x+3 = 4x-5 c) 2x-10 = 2x-10
(Followed by class discussion)
Break out rooms: Split elementary & secondary

14. Your Turn, Lens 1: Engaging Mathematical Argumentation
Lens 1: Does the task engage students in mathematical argumentation?
Alexa is training to bike 50 miles. During her first week of training she bikes 12 miles. During her second week she bikes 24 miles, and by her third week she bikes 36 miles. On what week does she bike close to 50 miles?
Alexa is training to bike 100 miles. During her first week of training she bikes 12 miles. On her fifth week she bikes 40 miles. Write an equation to represent her training progress and use it to determine on what week she will be able to bike 100 miles.

15. Your Turn, Lens 1: Engaging Mathematical Argumentation
Lens 1: Does the task engage students in mathematical argumentation?
The coordinates of the four vertices of figure ABCD are
A(4, 3), B(8, 3), C(4, 6) and D(8, 6). Is figure ABCD a rectangle?
The coordinates of the four vertices of parallelogram ABCD are
A(-4, -3), B(5, 6), C(8, 3) and D(-1, -6).
Determine the slopes and lengths of the sides to verify that it is a rectangle.
c. Does the task require students to write or otherwise record their chain or reasoning to show the result is true?
The coordinates of the four vertices of parallelogram ABCD are
A(-4, -3), B(5, 6), C(8, 3) and D(-1, -6)
After some calculations, Jasmine has classified this as a rectangle.
Use your knowledge of parallelograms to create a mathematical argument that will verify her conclusion.

16. Break Time

17. Lenses for thinking about tasks
Lens 1: (Engagement)
Does the task engage students in mathematical argumentation? How can it be modified?
Lens 2: (Student Purpose)
What do you want the students to learn from the mathematical argumentation task?
Lens 3: (Teacher Purpose)
What will you learn about students by using the mathematical argumentation task?

18. Lens 2: Purposes for Argumentation Tasks
What do you want the students to learn from the argumentation task?
Does the task help students produce better arguments?
Does the task help students develop conceptual understanding?
Does the task help students analyze concepts through collaborative work?
Does the task help students to mathematize contextualized problems and interpret the meanings of solutions?
Does the task help students to understand how representations communicate mathematical ideas AND develop proficiency working across multiple representations?
Reference the Lens 2 handout: Some Purposes for Argumentation Tasks to Support Student Learning

19. Your Turn… Is it a ¼? & DJ problems - Lens 2
Lens 2: What do you want the students to learn from the argumentation task?
Laura says that ¼ of the rectangle is shaded. Do you think she is correct? Write an argument to support your answer.
In preparation for the Prom, students are researching the costs of two local DJ companies. Music Makers charges a fee of $200 and an additional $175 per hour. Dance Partners does not charge an initial fee, but charges $225 per hour. Which company would be more cost effective for the prom committee?
Write a mathematical argument to support your decision.
Break Out into Elementary & Secondary Groups

20. Lens 2: Purposes for Argumentation Tasks
What do you want the students to learn from argumentation task?
DO NOT Solve.
Discuss how you would solve each of the following problems. Support your ideas by explaining your mathematical thinking.
a) = ____ b) 500 ÷ m = c) 25 x 10 x 4 = _____
Discuss solutions to each equation. Support your thoughts with a mathematical argument.
a) 3x+5 = 2x-6 b) 4x+3 = 4x-5 c) 2x-10 = 2x-10
These may be just handouts – use for reference if necessary

21. Lens 2: Purposes for Argumentation Tasks
Lens 2: What do you want the students to learn from argumentation task?
Alexa is training to bike 50 miles. During her first week of training she bikes 12 miles. During her second week she bikes 24 miles, and by her third week she bikes 36 miles. If Alexa continues with the same biking pattern each week, will she be able to bike 50 miles by week 4? Write a mathematical argument to support your reasoning.
Alexa is training to bike 100 miles. During her first week of training she bikes 12 miles. On her fifth week she bikes 40 miles. If she continues to advance in this same pattern, on what week will she be able to bike 100 miles? Write a mathematical argument to support your answer.

22. Lens 2: Purposes for Argumentation Tasks
Lens 2: What do you want the students to learn from argumentation task?
The coordinates of the four vertices of figure ABCD are
A(4, 3), B(8, 3), C(4, 6) and D(8, 6). Based on the differences between the coordinate points, Jasmine believes figure ABCD is a square. Do you agree with her? Write a mathematical argument to support your answer.
The coordinates of the four vertices of parallelogram ABCD are
A(-4, -3), B(5, 6), C(8, 3) and D(-1, -6).
After some calculations, Jasmine has classified this as a rectangle.
Use your knowledge of parallelograms to create a mathematical argument that will verify her conclusion.

23. Lenses for thinking about tasks
Lens 1: (Engagement)
Does the task engage students in mathematical argumentation? How can it be modified?
Lens 2: (Student Purpose)
What do you want the students to learn from the mathematical argumentation task?
Lens 3: (Teacher Purpose)
What will you learn about students by using the mathematical argumentation task?

24. Lens 3: Informing Instruction
When would I use this task? Why am I using it? What will I learn from it?
Would the task help introduce a concept?
Do students have prior knowledge about this topic?
Have students mastered the targeted skills or concepts?
Will the task help students to apply or connect their knowledge of one or more topics from a unit or lesson?
Can students effectively communicate their reasoning and make clear connections between their claims, warrants and evidence?

25. Is it a ¼? & DJ problems - Lens 3
Lens 3: What will I learn about students by using the argumentation task?
Laura says that ¼ of the rectangle is shaded. Do you think she is correct? Write an argument to support your answer.
In preparation for the Prom, students are researching the costs of two local DJ companies. Music Makers charges a fee of $200 and an additional $175 per hour. Dance Partners does not charge an initial fee, but charges $225 per hour. Which company would be more cost effective for the prom committee?
Write a mathematical argument to support your decision.

26. Lens 3: Informing Instruction
Lens 3: What will I learn about students by using the argumentation task?
DO NOT Solve.
Discuss how you would solve each of the following problems. Support your ideas by explaining your mathematical thinking.
a) = ____ b) 500 ÷ m = c) 25 x 10 x 4 = _____
Discuss solutions to each equation. Support your thoughts with a mathematical argument.
a) 3x+5 = 2x-6 b) 4x+3 = 4x-5 c) 2x-10 = 2x-10

27. Lens 3: Informing Instruction
Lens 3: What will I learn about students by using the argumentation task?
The coordinates of the four vertices of figure ABCD are
A(4, 3), B(8, 3), C(4, 6) and D(8, 6). Based on the differences between the coordinate points, Jasmine believes figure ABCD is a square. Do you agree with her? Write a mathematical argument to support your answer.
The coordinates of the four vertices of parallelogram ABCD are
A(-4, -3), B(5, 6), C(8, 3) and D(-1, -6).
After some calculations, Jasmine has classified this as a rectangle.
Use your knowledge of parallelograms to create a mathematical argument that will verify her conclusion.

28. Lens 3: Informing Instruction
Lens 3: What will I learn about students by using the argumentation task?
Alexa is training to bike 50 miles. During her first week of training she bikes 12 miles. During her second week she bikes 24 miles, and by her third week she bikes 36 miles. If Alexa continues with the same biking pattern each week, will she be able to bike 50 miles by week 4? Write a mathematical argument to support your reasoning.
Alexa is training to bike 100 miles. During her first week of training she bikes 12 miles. On her fifth week she bikes 40 miles. If she continues to advance in this same pattern, on what week will she be able to bike 100 miles? Write a mathematical argument to support your answer.

29. Tasks Alone Are Not Enough
The ways teachers use a task in the unique setting of their classroom at a particular moment in time determines whether or not that task is effective.
A task can have more than one focus. A teacher may use the same task for different purposes.
It takes time -- talk through samples; develop the practice of doing argumentation (verbal and written)
Asking students to generate a written argument for an assessment task is not equivalent to prompting students to engage in the practice of argumentation for the purpose of learning. These are related but distinct activities.
Problem Solving alone is not argumentation and vice-versa.
Mathematics might make the writing of an argument more difficult. Simple content may be intentionally selected to support focus on argumentation.

30. Task Alone Are Not Enough
Developing a Culture of Thinking in Mathematics…
“Children must be taught HOW to think not WHAT to think.”

31. Setting Up for the Afternoon
BRIDGING TO PRACTICE

32. Viewing through the three lenses, analyze your own task.
Viewing Your Own Tasks
Task Analysis:
Viewing through the three lenses, analyze your own task.
Lens 1: (Engagement)
Does the task engage students in mathematical argumentation? How can it be modified?
Lens 2: (Student Purpose)
What do you want the students to learn from the mathematical argumentation task?
Lens 3: (Teacher Purpose)
What do you want to learn about students by using the mathematical argumentation task?

33. Working Lunch
Review the three lenses over lunch and discuss the following:
which lens do you feel you consider consistently in your planning of instruction?
which lens do you feel doesn’t get enough consideration in your planning of instruction?

34. Bridging to Practice Group Work: Protocol
Examine the tasks that you brought using the lenses
New small groups into main room and breakout room
Remember to collect protocol work

35. Thank you for your feedback!
Survey
Thank you for your feedback!

36. Acknowledgements
This work was supported by a Math-Science Partnership Grant from the State Department of Education.
Bridging Math Practices was a collaborative project among UConn, Manchester Public Schools, Mansfield Public Schools, and Hartford Public Schools