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Project Reflection

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 Рbefore 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 Tuesday, December 16th, at midnight).  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 (“let the games begin”) 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?
  • Do you have any advice for future students in MAT 2071 about the project?

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.

Semester Project – Group Presentations: Description and Grading Criteria

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:

  • Briefly describe the rules of your puzzle (you can include an example). ¬†The class has had a chance to look at these puzzle before, so this serves as a reminder of the rules instead of a step-by-step training.
  • 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 may use a PowerPoint or other presentation, or simply give your talk in front of the blackboard.
  • 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 Tuesday, 12/9 and Thursday, 12/11. ¬†Your group must sign up for a presentation time before leaving class on 12/2.


Grading Criteria (20 points total)

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

_____ points (3 possible).  Puzzle description.  Describe basic rules of puzzle (example optional).

_____ points (5 possible).  Conjecture.  Conjecture is written down.  Explanation and example are provided.

_____ points (5 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)


Semester Project – Group Process Paper

UPDATE 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) Рcomplete details can be found on the Project Overview & Deliverables page.  The group paper assigned here forms a significant portion of the project.

Semester Project – Group Process Paper

In his essay A Mathematician‚Äôs Lament, Paul Lockhart says ‚ÄúA good problem is something you don‚Äôt know how to solve.‚ÄĚ This is quite different from most of the ‚Äúproblems‚ÄĚ that appear our mathematics education. ¬†In the past weeks, you‚Äôve all spent some time individually and in groups working on such a problem (the MIU game, Bridges and Walking Tours, and Mutilated Checkerboards puzzles).

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

  • Description of your puzzle, in your own words.
  • An account of working on your puzzle as a group, from playing with the puzzle 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 – ‚Äúwhat was it like‚ÄĚ?.
  • A statement of your group‚Äôs 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‚Äôve 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.

Details:  This assignment is due Tuesday, December 2nd.  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.


OpenLab #9: Proof Journal

Your assignment for the next week is to try to prove the conjecture that your group created in class on Thursday, 11/6/14.  You must spend at least 90 minutes working on this (including 30 minutes in class on Tuesday, 11/11).  Trying to prove something can consist of many different activities, such as the following (you do NOT have to do all of these things Рyou can choose how to spend your time Рthey 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 Рan 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 Рwere you frustrated/confused/happy/depressed? Why? Did your mood change along the way?

Assignment (Due Thursday, 11/13/14):  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 or other work you did along the way.

OpenLab #8: Lockhart’s Lament

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 Thursday, 11/6/14). 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 or question about your game, and bring it with you to class on Thursday 11/6 (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. ¬†Think about¬†the game you¬†worked on last week (the MIU game, the bridges and walking tours game, or the mutilated checkerboards game). ¬†Each of these games 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 Thursday 11/6 (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:
Question: Is the corner the best move, or the center?
Conjecture: The person who goes first always wins.
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.”

Week 9 Assignments

Written work, Due Tuesday, November 4th, in class:
Chapter 7 p129: 5, 9, 10, 12
**Chapter 8 p143: 3, 7, 18, 19
      **Chapter 8 has been extended to Tuesday, November 11th
OpenLab¬†–¬†OpenLab #8 due Thursday, November 6th¬†at the start of class

Class work: There will be group & individual work completed & submitted in class on Thursday 10/30/14, which will count towards your “Project” grade.

OpenLab #7: Let the games begin

Hi everyone,

Based on your responses to OpenLab #6,¬†I have assigned each of you a game to work on (everyone got their first or second choice). ¬†Check the “Who is doing what” list below to see which game you will be working on. ¬†Then follow the appropriate link from the list below to view your assignment (detailed instructions are provided for each game).

Prof. Reitz

Links to assignments (Due TUESDAY, OCTOBER 28th – end of day):

Who is doing what:

Chiu,SinFong Bridges and Walking Tours
Conyers,Sarah Bridges and Walking Tours
Felamon,Marina MIU Game
Garcia,Yanira Bridges and Walking Tours
Hamza,Syed Mutilated Checkerboards
Jones,Stacy Mutilated Checkerboards
Kamath, Neil Bridges and Walking Tours
Laing,Latina Mutilated Checkerboards
Lee,Victor MIU Game
M Rafeek, Rushdha Mutilated Checkerboards
Mongo,Joseph Mutilated Checkerboards
Perez,Leonardo MIU Game
Rivera,Julia Mutilated Checkerboards
Ruiz,Joseph Mutilated Checkerboards
Sao,Sidney Mutilated Checkerboards
Shati,Farjana MIU Game
Sun,Jian Mutilated Checkerboards

OpenLab #6: Choose a game!

What is it that mathematicians really do? ¬†They prove theorems, it’s true – but that’s just¬†the end result. ¬†Along the way comes the fun part – playing around with a problem, trying to see if you¬†can understand the rules of the game, and find some kind of¬†pattern that “makes the whole thing make sense”. ¬†Over the next few weeks, you will be working individually and in groups on a puzzle or game. ¬†Of all the things¬†you do in this class, this work will be most similar to the work that real mathematicians do.

OpenLab #6: Due this Thursday 10/16

Read through the descriptions of the three puzzles below. ¬†Then respond to this comment, ranking them in order of preference. ¬†That’s all you have to do.

For example, you might respond like this:

First choice: Mutilated Checkerboards
Second choice: Bridges and Walking Tours
Third choice: MIU Game

NOTE: This project is about the process of playing around with a problem when you do not know what the answer is (or even what the problem is, really).  BUT all of these puzzles are well-documented on the internet.  I request that you do not Google them, or look at any other resources related to them, until this project is over.


  • Read the puzzle¬†descriptions below. ¬†Do NOT Google these topics for more information.
  • If you¬†are already familiar with one of these puzzles, pick¬†another one as your first choice.
  • Puzzles will be assigned next week, with more information and specific instructions.
  • You do NOT need to solve these puzzles for now.

The Puzzles

Puzzle 1: The MIU Game

We are going to play a game with strings of symbols.  This game was invented by a man named Douglas Hofstadter and found in his book Gödel, Escher, Bach. Here are the rules:

Suppose there are the symbols ‘M’, ‘I’, and ‘U’, which can be combined to produce strings of symbols called “words”, like MUI or MIUUU. In the MIU game, we always start with the “axiomatic” word MI, and transform it using the following four rules, to obtain some “goal” word. ¬†The rules state that you may:

  1. Add a U to the end of any string ending in I. For example: MI to MIU, or MUUII to MUUIIU.
  2. Double any string after the M (that is, change Mx, to Mxx, where ‘x’ represents any string of symbols). For example: MIU to MIUIU
  3. Replace any III with a U. For example: MUIIIU to MUUU
  4. Remove any UU. For example: MUUU to MU

Goal Word: MIIU

Start with MI (axiomatic word)
MI to MII (rule 1)
MII to MIIU (rule 2)



Puzzle 2: Mutilated Checkerboards

We are going to play a game with checkerboards and dominos. ¬†So that we all have the same picture in our heads, a checkerboard is an 8×8 grid of squares (64 squares total), colored alternatingly black and white:

A domino is a 1×2 block, which is exactly the size of two squares on the checkerboard (the dots on the domino do not mean anything in this game):

Your goal is to cover the checkerboard with dominos, according to the following rules:

  1. You can use as many dominos as you wish.
  2. Dominos can be arranged horizontally or vertically, as long as each domino covers exactly two squares on the checkerboard.
  3. Dominos cannot overlap each other, and cannot extend off the edge of the board.

EXAMPLE 1:  Can you cover the checkerboard with dominos?

EXAMPLE 2: What if we remove one of the corner squares from the checkerboard (a “mutilated checkerboard”) – now can you cover it with dominos?


Puzzle 3: Bridges and Walking Tours

We are going to play a game creating walking tours of cities with bridges.  We begin in the city of King’s Mountain, which is built on four land masses Рboth shores of a river and two islands in midstream Рconnected by a total of seven bridges (shown in green).

EXAMPLE 1:  Can you create a walking tour of the city that crosses every bridge exactly once?  You can begin anywhere you like, and end anywhere you like, as long as you cross each bridge just once.