Hi everyone,

Attached are the final drafts of each group paper – excellent work, everyone!

Best regards,

Prof. Reitz

ismael_jeron_hanan_proofs-and-logic-group-paper

Hi everyone,

Attached are the final drafts of each group paper – excellent work, everyone!

Best regards,

Prof. Reitz

ismael_jeron_hanan_proofs-and-logic-group-paper

This assignment is the final deliverable for your project (worth 10 points). It is an individual, not a group, assignment and should be submitted **by email, not on the OpenLab.**

The Semester Project consisted of a number of related activities and assignments – before you begin writing, please take a look at the list and click each of the links to remind yourself of all the parts of the project.

**Assignment**** (Due before your final exam – Tuesday, December 20th)**. SUBMIT BY EMAIL, NOT ON THE OPENLAB. This is your chance to reflect on the Semester Project, and to describe your contributions to the major group activities. Please respond to all of the following.

Part 1 (200 words minimum). Write one or two paragraphs reflecting on the Semester Project, from the initial assignment (“OpenLab #5: Bridges and Walking Tours“) through to the final presentations. You can use the following questions to guide your writing if you wish (or you can respond in whatever way you wish):

- What was your overall impression of the project? What did you enjoy the most? Least?
- Do you feel that the project enhanced your experience of the class? Was the project related to the course material?
- How did you feel during the process? What was the best moment? The worst?
- Was the project scaffolded appropriately? Did you need more or different support for any of the assignments? Do you have any suggestions for making the project more effective?

Part 2. Describe your personal contributions to the final two major group assignments, the group paper and the group presentation. Do you think that you were an equal contributor in your group? Be as specific as you can.

The last significant group assignment for your semester project is a group presentation (there will be one more individual assignment, a reflection on the process). I’ll put the details here, followed by an outline of the grading criteria (the presentation is worth 20 points total).

**Semester Project – Group Presentation**

This is your chance to share your group’s work with the rest of the class. Each group will give a 5-8 minute presentation, including the following items:

- State your conjecture (this should be written down, either on a slide or on the board). Give an explanation, and an example to demonstrate your conjecture.
- If you were able to prove your conjecture, give a proof. If not, describe briefly some of the ideas you had and strategies you tried while trying to prove it.
- Give the class at least one puzzle to work on on their own – a challenge!
- Give the audience a chance to ask questions (either during the presentation, or after).

Keep in mind the following:

- You must include some kind of slides (you may also put work on the board): PowerPoint, Google Slides, Prezi.com, LaTeX Beamer, or other.
- You may decide as a group how to divide up the work, but
*each group member must present something*to class. - Be aware that you will be asked at a later time to describe your own specific contributions as well as those of each group member.
- Presentations will be given at the beginning of class on Thursday, 12/1 and Tuesday, 12/6.
*Your group must sign up for a presentation time before leaving class on 11/20.*

**Grading Criteria (20 points total)**

_____ points (4 possible). Basics.* Stay within time limits (5-8 minutes). All group members participate.*

_____ points (6 possible). Conjecture.* Conjecture is written down. Explanation and example are provided.*

_____ points (7 possible). Proof of conjecture or proof process description.

_____ points (3 possible). Challenge the class. *At least one puzzle is given for the class to work on on their own.*

** **

**____ points TOTAL (20 possible)**

Hi everyone,

The group process paper will be worth 35 points towards your Project grade. I will be filling out the sheet below for each paper submitted. Please let me know if you have any questions.

Best,

Prof. Reitz

**Semester Project – Puzzle Process Paper
**

_____ points (3 possible). Basics/formatting.* Length (1500 words required). Group members names. Semester/Date/Course.*

_____ points (2 possible). Puzzle description. *Description given in own words, demonstrates understanding of puzzle mechanics.*

_____ points (16 possible). Proof process narrative.

*_____ points (4 possible). Shows progress across various stages of the project. *

*_____ points (4 possible). Includes all participating members of the group. *

*_____ points (4 possible). Includes objective facts (“what we did”) as well as experience (“how it felt, what it was like”). *

*_____ points (4 possible). Tells a story.*

_____ points (5 possible). Conjecture.

*_____ points (3 possible). State your group’s conjecture.*

*_____ points (2 possible). Proof or disproof of conjecture. If no proof or disproof was obtained, these points can be earned by clear explanation of proof process in the preceding account.*

_____ points (9 possible). Images (3 points each). *Original or clearly attributed. Includes caption. Connection to puzzle/process is evident. *

**____ points TOTAL (35 possible)**

**Assignment.** Your goal for today is to *refine* the conjecture you decided on during your last class meeting. Some things to consider:

**Specificity:**The conjecture should be stated clearly. It should include all information necessary to be understood by someone who is familiar with graph theory terms (vertex, edges, paths) and familiar with the assignment (walking tours).*A reader should be able to tell from the statement whether a conjecture applies to a given drawing or not.***Generality:**Your conjecture should apply to more than just a single specific graph (it can apply to a collection of similar graphs, for example, as long as you describe exactly what types of graphs you are considering).**Drawing:**You can create a drawing to accompany your conjecture, but your conjecture should be understandable without the picture.- You can revise your conjecture as a group if you wish – but try to come up with something similar.
- You can add additional clarification to your conjecture.
- You can extend your conjecture to include more types of graphs.

GROUP CONJECTURES SUBMITTED AT END OF CLASS TODAY:

In his essay *A Mathematician’s Lament*, Paul Lockhart says “A good problem is something you don’t know how to solve.” This is quite different from most of the “problems” that appear in our mathematics education. In the past weeks, you’ve all spent some time individually and in groups working on such problems, in the context of graph theory (“Bridges and Walking Tours”).

As a group, write an account of your experiences working on your puzzle/problem. You should include the following elements:

- Description of the Bridges and Walking Tours problem, in your own words.
- An account of working on your problem as a group, from playing with the problem to formulating and perhaps proving a conjecture. What did your group do/think/feel? You can include examples of puzzles and solutions if you wish, as well as work by individual group members completed outside the group (both optional).
*Your goal is not to go over every detail, but to tell a story that your readers will enjoy – “what was it like”?.* - A statement of your group’s chosen conjecture, and a proof (or disproof) of the conjecture.
- At least three images (more if you wish). They can include images of puzzles you’ve created or solutions, but you can also be creative with images or photos related to your puzzle, your group or your story in some way. Each image should have a caption describing.
*NOTE: You may freely use your own drawings, images or photos. If you wish to use photos from another source, they must be from a legal source (for example, Creative Commons licensed, with proper attribution – the library or your professor can help with this).* - Basic details: the names of all group members, the date, course and section numbers, and your professor’s name.

**I will be meeting with each group next Thursday, November 10th, in class.** Please be in touch with your other group members before then! Be prepared to discuss your progress so far – at the very least, you should be able to describe how you are dividing up the work of the paper among your group.

**The first draft of this assignment is due in class on Thursday, November 17.** Each group should submit one paper, of no less than 1500 words. You may decide as a group how to divide up the work. Be aware that you will be asked at a later time to describe your own specific contributions as well as those of each group member.

**The final draft of this assignment is due in class on Thursday, December 1.**

**REGARDING SEMESTER PROJECT: ** As you may recall from the Course Description, the semester project is worth 10% of your overall grade. The project consists of a number of interrelated activities (many of which have already been completed) – complete details can be found on the Project Overview & Deliverables page. The group paper assigned here forms a significant portion of the project.

Your assignment for the next week is to try to prove the conjecture that your group created in class on Tuesday, 10/14/15. **You must spend at least 90 minutes working on this.** Trying to prove something can consist of many different activities, such as the following (you do NOT have to do all of these things – you can choose how to spend your time – they are provided for inspiration only).

- coming up with ideas, and testing them out (for example, by creating puzzles and trying to solve them)
- trying to understand what the conjecture says
- trying to solve puzzles that other people created
- trying to create puzzles (and solve them yourself)
- communicating with other members of your group (talking, emailing, etc.)
- trying to write down a proof
- other stuff…

As you work, keep track of what you are doing, thinking, and feeling (this is **metacognition** – an idea that discussed way back in OpenLab #2). What did you do during the time you spent? Did you create any puzzles? Did you solve puzzles? Did you change your mind about whether the conjecture is true or false? Did you have any new ideas about how to prove the conjecture? Did you have any ideas that you gave up on? How did you feel as you worked – were you frustrated/confused/happy/depressed? Why? Did your mood change along the way?

**Assignment (Due Thursday, 11/3/15)**: Submit a journal of your efforts in the comments below. Your response should be **at least 300 words**. Describe what you did during the 90 minutes you worked, and express in some way what you were thinking and feeling during the process. Your response can include puzzles (use sketchtoy.com) or other work you did along the way.

**Extra Credit**. Respond to a fellow student’s comment. Did you do similar things? Different things? Do you have any suggestions for them? Be kind.

GROUP CONJECTURES:

*NOTE: As a component of OpenLab #5, each person should bring to class a conjecture or question about the Bridges and Walking Tours game.*

**Group Activity (30 min).** Get into your groups, arrange your chairs in a circle, and take 30 minutes to:

1. Each person should share their conjecture with the group. For each conjecture, the group should decide if they think it is true or false (or don’t know). The group should record their conclusion for each conjecture.

2. Choose one conjecture (or create a new one) to focus on as a group. Your goal for the next few weeks will be to try to prove or disprove this conjecture. Come up with several ideas about how you might prove it.

**Group work due after 30 minutes:** Each group will hand in a sheet of paper with the names of the group members, the date, and the following:

– Each member’s conjecture, along with a brief description of what the group thinks – is it true or false?

– Be sure to clearly indicate which of the conjectures the group has chosen to work on – or, if you have created a new conjecture to work on as a group, include that as well.

– Two different ideas about how you might try to prove the chosen conjecture.

**Reflection:** To be completed individually after group work is complete, and submitted on paper with your names and the date. Take 5 minutes to write on the following prompt:

Briefly reflect on the process of working in a group by responding to each of these points:

1. Describe something you learned.

2. Describe something you contributed to the group.

3. How did today’s work change your understanding of your assigned game?

**Group Activity (25 min).** Get into your groups (group assignments appear below), arrange your chairs in a circle, and take 25 minutes to:

1. Share your responses to OpenLab #4. Compare your answers to the given examples (do you all agree on the solutions?). Now share your own puzzles with the group, and discuss.

2. Create three new puzzles, as tricky as possible (try to stump Prof. Reitz!).

**Group work due after 25 minutes: **Each group will hand in a sheet of paper with the names of the groups members, the date, and the three new puzzles created by the group. You do not need to submit solutions, but you do need to be able to solve the puzzles on request.

**Reflection:** To be completed individually after group work is complete, and submitted on paper with your name and the date. Take 5 minutes to write on the following prompt:

Briefly reflect on the process of working in a group by responding to each of these points:

1. Describe something you learned.

2. Describe something you contributed to the group.

3. How did today’s work change your understanding of your assigned game?

**Group 1:** Hanan, Ismail, Jeron

**Group 2:** Jose, Violanda, Kevin, Carter

**Group 3:** Tyniqua, Sonam, Ahmad

**Group 4:** Gary,Leonardo, Armando

In 2002, a mathematician named Paul Lockhart wrote an essay called “A Mathematician’s Lament,” a passionate criticism of mathematics education in America. It has become widely known among mathematicians and mathematics educators – not everyone agrees with everything he says (though many do), but everyone seems to have something to say about “Lockhart’s Lament,” as it is called. For this week’s assignment, you will read a short excerpt (three pages) from his essay and respond to the prompts below.

**Assignment (Due Friday, 10/14/16)**. Your assignment has three parts:

**First, read** the section titled “Mathematics and Culture” (pages 3-5) in Lockhart’s essay, (click here). *If you’re interested, I encourage you to read more, starting at the beginning – but this is not required.*

**Second, write a response** to what you read and post it in the comments below. Your response should be **at least 300 words.** Your response should represent your own thoughts and opinions on what you read, and can include responses to any or all of the following:

- What is one thing that you agree with in the reading? Explain why.
- What is one thing that you do not agree with? Explain.
- Choose one quote that you think stands out in the reading. Give the quote, and explain why you chose it.
- Have you ever had an experience of mathematics as art?
- On page 5, Lockhart describes mathematics in schools today as “heartbreaking”. What do you think he means? Do you agree? How do your own math experiences in school compare to his description?

**Third**, and most important, I want you to **write down a conjecture about the Bridges and Walking Tours game, and bring it with you to class on Friday 10/14 (do NOT post it here)**. Consider Lockhart’s example of a triangle drawn inside a rectangle. He described the process of playing around with this picture, until he arrives at the basic idea for calculating the area of a triangle. He contrasts this with a traditional math class, in which the formula is given to students without providing them any opportunity to explore the problem on their own. The bridges and walking tours game is a little like the triangle-rectangle picture – it’s fun to play around with, but you may not be sure what the point is. You’ve had a chance to play with it a bit, and try some different challenges. Now what? Your job is write down a conjecture (a guess!) or a question about your game. If you could have one question answered about your game, what would it be? If you wanted to be a master of your game, and be able to solve any challenge that was given to you, what would you need to know? **Write down a conjecture or question about your game, and bring it with you to class on Friday 10/14 (do NOT post it here).**

Here is an example: Let’s imagine that you have just been introduced to the game Tic-Tac-Toe. After playing it for a while, you might come up with one of the following:

Conjecture: The person who goes first always wins.

Question: Is the corner the best move, or the center?

Conjecture: It’s impossible to win, no matter who goes first.

ps. Paul Lockhart retired from being a first-rate research mathematician in order to teach math at a private elementary school here in Brooklyn, Saint Ann’s School, where he says “I have happily been subversively teaching mathematics (the real thing) since 2000.”

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