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Write Proofs!
Using the Logic in Games to Develop
Students’ Proof-Like Reasoning and Communication
email: [email protected] http://lucywestpd.com phone: 212-­‐233-­‐0419 cell: 917-­‐494-­‐1606 fax: 212-­‐608-­‐0714 Write Proofs! Using the Logic in Games to Develop Students’ Proof-­‐like Reasoning & Communication 1.  What is a winning strategy? Play Tic-­‐Tac-­‐Toe with a partner. Ask yourself: 2.  How might you convince someone that this is a winning strategy? 3.  Write your argument and share with a neighbor. —  1.  Welcome & Introductions 2.  A Short History of Our Collaboration 3.  Using Games to Develop Proof-­‐like Reasoning 4.  Impact of Games on Student Reasoning and Communication 5.  Questions Presenters: Metamorphosis TLC, New York City —  Antonia Cameron, Director of Professional Development PS 230, Brooklyn, New York —  Karine Kelley, Grade 4 teacher & math teacher leader —  Lauren O’Neill, Grade 3 teacher & math teacher leader —  Desiree Carver-­‐Thomas, Grade 4 teacher & math teacher leader PS 686, Brooklyn, New York —  Melissa Singer, Math Co-­‐teacher, Grades 4 & 5 The journey began at a public school, PS 230. PS 230 is a public pre-­‐K-­‐5 school in Brooklyn, New York. •  Enrolls about 1,400 students. •  27 different languages are spoken at PS 230. •  64% of the students have free lunch. •  34.5% of the students are ELLs. •  14% of the students receive special education services. •  12% of the students are in a gifted and talented program. •  Coach brought in to develop teachers’ mathematical content and pedagogical content knowledge •  Created teacher leaders in the process •  Teacher leaders co-­‐facilitated teaching learning communities What are teaching learning communities? What are teaching learning communities? •  Full-­‐day situated-­‐based learning •  Focused on specific mathematical content or processes •  Co-­‐facilitated by me and the teacher leaders •  Public learning: teacher leaders open their practice to the scrutiny of others •  Teachers from around NYC participate Our focus for the past two years has been on developing students’ oral and written communication. Where did we begin our work together? Our journey began with some questions. These were linked to the CCSS mathematical practices. 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 repeated reasoning. How does a student’s ability to write in mathematics develop? What role does mathematical reasoning play in the development of oral and written communication? Is there a learning trajectory for this development? What kinds of tasks are critical to the development of mathematical reasoning and communication? What constitutes proof in the elementary math classroom? We used a broader definition of “proof” in our work with children. We think of proof as a socially constructed process whose norms for what constitutes ‘acceptable’ arguments have fluid boundaries that are, to an extent, unique to the community it serves. Stylianou, Blanton, & Knuth, Teaching and Learning Proof Across the Grades, A K-­‐16 Perspective 1.  What kind of reasoning is proof-­‐like? 2.  What role do conjectures play in the development of students’ proof-­‐ like reasoning? 3.  How might we use students’ conjectures to develop their ability to create logical proof-­‐like arguments? Overview of the Game Sequence and Teaching Structure Sequence of Games Grade 3 4 5 Game Tic-­‐Tac-­‐Toe Nim, Jr. Nim Nim Babylon Babylon Othello The Basic Structure & Varia6ons Play •  Teacher plays class •  Students play each other in teams •  Keep track of strategies in notebooks •  Choose a strategy and try it out •  Follow a student’s directions exactly Revise •  Try to lose Share •  Gallery Walk, Congress •  Reflection (Independent writing, finish a game using what you’ve learned) Tic-­‐Tac-­‐Toe You played Tic-­‐
Tac-­‐Toe with a partner. Question: How might this game develop proof-­‐like reasoning? One Student’s Wri6ng on Tic-­‐Tac-­‐Toe The Basic Structure & Varia6ons Play •  Teacher plays class •  Students play each other in teams •  Keep track of strategies in notebooks •  Choose a strategy and try it out •  Follow a student’s directions exactly Revise •  Try to lose Share •  Gallery Walk, Congress •  Reflection (Independent writing, finish a game using what you’ve learned) —  The Goal: Force your opponent to remove the last tile. —  How to Play: On each turn, a partner removes one or more tiles from a single row. 1. 
Play a few rounds of Nim, Jr. with a partner. 2.  As you play, think about these questions: •  Is there a winning strategy? •  If yes, what is it? •  How would you convince someone else of your strategy? 3. 
Write a conjecture about a winning strategy. A typical conjecture is If you go first, you will always win. Do you agree with this conjecture? Disagree? Kids need time to • 
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Explore Try out each other’s conjectures. Prove/disprove conjectures. Guess what? Children need time to explore; so do adults. In our TLC, teachers spent the morning exploring Nim Jr. and Nim, thinking about the questions we just asked you to consider. They developed conjectures. They thought about proving their conjectures. Here’s a sample of writing. Read the writing. Do you agree with what’s written? Why or why not? How does this writing approximate ‘proof’? Insert teacher writing from TLC
CCSS Mathema6cal Prac6ces in Ac6on What does proof-­‐like reasoning sound like in student talk? CCSS Mathema6cal Prac6ces in Ac6on One Student’s Strategy: If Player 1 takes the top tile, then they will win. What’s critical to develop proof-­‐like reasoning: • 
Kids are given the time to try out a conjecture. What’s critical to develop proof-­‐like reasoning: • 
Children choose the direction of their own learning. What’s critical to develop proof-­‐like reasoning: • 
New conjectures arise from exploration. Maya realizes she can leave a 2 by 2 arrangement of tiles and win the game. What’s critical to develop proof-­‐like reasoning: • 
Big ideas also arise from exploration: A big ah-­‐ha moment for many students: At what point do you know you’ve lost? What’s critical to develop proof-­‐like reasoning: • 
Different kinds of reasoning and uses of language emerge. Use of if-­‐then reasoning/language: Maya: if a player sets this up, then I will win. Recipe-­‐like reasoning/language: Nina: Player 1 takes the top tile and player 2 takes one of the bottom tiles, then players 2 has lost. The Goal: Force your opponent to remove the last tile. How to Play: On each turn, a partner removes one or more tiles from a single row. Question: “Can you use your strategy from Nim, Jr. to help you win the game of Nim?” If-­‐then play-­‐by-­‐play recipe Insert student writing samples
The Basic Structure & Varia6ons Play Revise Share •  Teacher plays class •  Students play each other in teams •  Keep track of strategies in notebooks •  Choose a strategy and try it out •  Follow a student’s directions exactly •  New ideas and conjectures emerged •  Generalizations also started to happen •  Gallery Walk, Congress •  Reflection (Independent writing, finish a game using what you’ve learned) Think before moving Consider skill-­‐level of opponent If-­‐then Generaliza
tions Account for different outcomes Using Games to Develop Proof-­‐like Reasoning And then what …? Using Games to Develop Proof-­‐like Reasoning How did our work with games affect student oral and written communication in math? Using Games to Develop Proof-­‐like Reasoning Students began constructing conjectures and testing them out. As part of this process, they were willing to revise their conjectures and writing. Clarity became key. Using Games to Develop Proof-­‐like Reasoning Persistence became critical to finding all the possible solutions to a particular problem. Insert student writing samples
Using Games to Develop Proof-­‐like Reasoning Students were willing to write mathematical statements to disprove other student’s thinking. “You think this; I disagree. Let me PROVE it.” By exploring, inquiring, and analyzing within the context of playing games, students make conditional assertions, name key moves or situations and, ultimately, make absolute statements about the game (Olmstead 2007; Reid 202a). P. Janelle McFeetors and Ralph T. Mason Learning Deductive Reasoning through Games of Logic Games create a social context in which students engage in convincing each other by offering claims of reasoning that approximate the structure of mathematical proving. (Reid 202b) P. Janelle McFeetors and Ralph T. Mason Learning Deductive Reasoning through Games of Logic Students experiences with “proof” are rooted in their own inquiry process, which has the potential to help them develop a more positive affect toward mathematics. P. Janelle McFeetors and Ralph T. Mason Learning Deductive Reasoning through Games of Logic Thanks for coming to our workshop! Powerpoint available at: [email protected]