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https://2ndbook.org/wiki/File:Collatz_formal.pdf
https://2ndbook.org/wiki/File:Collatz_formal.pdf
----If someone were to prove the Collatz conjecture, here are some Socratic questions that could stimulate deep reflection and discussion:
----


## '''What did we assume to be true in the proof?'''
=== One-Week Quest: Understanding and Exploring the Collatz Conjecture ===
##* What things did we take for granted when proving it? Could there be other ideas that would lead us to different results?
## '''How does the proof look at even and odd numbers?'''
##* Why is it important to treat even and odd numbers differently in the Collatz sequence? How does this change the way the proof works?
## '''What does the proof tell us about math problems?'''
##* If the Collatz Conjecture is true, what does that tell us about other math problems that seem simple but are really hard to solve?
## '''What role did computers play in proving the conjecture?'''
##* How did using computers help in solving the Collatz Conjecture? Does this change the way we think about math proofs—are computers just as important as thinking with paper and pencil?
## '''What new questions does the proof bring up?'''
##* Now that the Collatz Conjecture is proven, what new questions or ideas can we explore? Did it help us notice other patterns in math?
## '''Can this proof help solve other math problems?'''
##* Are there other problems that are similar to the Collatz Conjecture? Could the ideas from this proof help us solve them too?
## '''What does the proof teach us about patterns in math?'''
##* Why does the Collatz sequence seem so strange and unpredictable, but still turn out to have a solution? What does this teach us about finding patterns in math?
## '''What does the proof mean for the bigger picture of math?'''
##* Does proving the Collatz Conjecture change how we think about math and what is possible to know? How does this affect how we search for answers in math?
## '''How do we know the proof is correct?'''
##* What steps do we take to make sure the proof is really right? How do we double-check to avoid mistakes in the reasoning?
## '''Does this proof change how we solve math problems?'''
#* Now that we have proven the Collatz Conjecture, does it change the way we should approach other unsolved math problems? Can we use the same techniques for other problems?


These questions focus on probing deeper into the implications, methods, and broader context of the proof of the Collatz conjecture, encouraging reflection on both the technical and philosophical aspects of the discovery.
==== Overview: ====
This week-long quest will guide students through the fascinating world of the Collatz Conjecture. They'll learn the background, explore the logic behind the conjecture, and engage in Socratic discussions about its implications in mathematics. By the end of the quest, students will have a deeper understanding of mathematical problem-solving, the role of computers in proofs, and the philosophical questions surrounding mathematical discoveries.
 
==== '''Day 1: Introduction to the Collatz Conjecture''' ====
 
* '''Objective''': Understand the Collatz Conjecture and its history.
* '''Activities''':
*# '''Watch a Video''': Begin by watching the introductory video on the Collatz Conjecture: Collatz Conjecture Video.
*# '''Read a Summary''': Briefly read the description of the Collatz Conjecture (provided on 2ndbook.org).
*# '''Discussion''': What do you think makes the Collatz Conjecture so interesting? How do you feel about the idea of a simple problem remaining unsolved for so long?
* '''Reflection Question''': What do you find most intriguing about the Collatz Conjecture?
 
==== '''Day 2: Understanding the Collatz Sequence''' ====
 
* '''Objective''': Learn how the Collatz sequence works and examine the behavior of even and odd numbers.
* '''Activities''':
*# '''Create Collatz Sequences''': Using various starting numbers, create Collatz sequences by following the rules: If the number is even, divide by 2; if odd, multiply by 3 and add 1.
*# '''Chart Sequences''': Track and chart the sequences for numbers 1–20. Look for any patterns or interesting observations.
* '''Reflection Question''': Why do you think the sequence behaves unpredictably at first, but always ends in 1?
 
==== '''Day 3: The Role of Computers in Proofs''' ====
 
* '''Objective''': Understand how computers contribute to proving the Collatz Conjecture.
* '''Activities''':
*# '''Read about Computational Contributions''': Explore how computers have been used to check millions of numbers to see if they follow the Collatz pattern.
*# '''Debate''': Have a class debate on the importance of computational methods in proving mathematical conjectures. Do you think that computers are as important as human reasoning?
* '''Reflection Question''': How does using a computer to check the sequences affect how we think about math proofs?
 
==== '''Day 4: Socratic Discussion – Assumptions and Mathematical Exploration''' ====
 
* '''Objective''': Engage in a Socratic discussion about the assumptions behind the Collatz proof.
* '''Activities''':
*# '''Socratic Questioning''': Discuss the following question: "What did we assume to be true in the proof of the Collatz Conjecture? Could there be other assumptions that might lead to different results?"
*# '''Group Discussion''': Break into small groups and discuss potential alternative assumptions in mathematics that could lead to different results.
* '''Reflection Question''': How can assumptions influence the outcomes of mathematical proofs?
 
==== '''Day 5: New Questions and Insights''' ====
 
* '''Objective''': Investigate what new questions arise now that the Collatz Conjecture might be proven.
* '''Activities''':
*# '''Group Brainstorming''': In small groups, brainstorm what new questions the proof of the Collatz Conjecture might open up. For example: Could similar patterns be found in other number sequences?
*# '''Class Presentation''': Have each group share their ideas about the new questions and insights.
* '''Reflection Question''': What new patterns or connections can we explore now that the Collatz Conjecture is proven?
 
==== '''Day 6: The Philosophical Implications of the Collatz Conjecture''' ====
 
* '''Objective''': Reflect on the philosophical implications of proving a long-standing mathematical puzzle.
* '''Activities''':
*# '''Socratic Discussion''': Discuss the question: "What does the proof of the Collatz Conjecture mean for the bigger picture of math? Does it change how we think about what is knowable in mathematics?"
*# '''Group Reflection''': How do you think the proof of a long-unsolved problem like the Collatz Conjecture affects our understanding of truth in math?
* '''Reflection Question''': Does proving the Collatz Conjecture change how we think about other unsolved problems in math?
 
==== '''Day 7: Final Project and Reflection''' ====
 
* '''Objective''': Synthesize your learning and reflect on the journey.
* '''Activities''':
*# '''Create a Presentation''': Create a short presentation about your journey through the Collatz Conjecture this week. Highlight the key learnings, new questions, and philosophical insights you’ve gained.
*# '''Class Reflection''': Share your reflections with the class and discuss how learning about the Collatz Conjecture has changed the way you think about math.
* '''Reflection Question''': What do you now think about the nature of mathematical problems? What new questions are you curious to explore?
 
----
 
=== '''Assessment:''' ===
Students will be assessed based on:
 
* Participation in discussions and Socratic questioning.
* Accuracy and insight in creating Collatz sequences and charts.
* Creativity and depth in brainstorming new questions and philosophical reflections.
* Quality and clarity of their final presentation and reflections.
 
This quest will engage students with both the technical and philosophical aspects of mathematics, providing them with a deeper appreciation of how even simple problems can lead to profound discoveries.
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