# Category: OpenLab Assignments

Your assignment for the coming week+ is to try to prove the conjecture that your group created in class on Thursday, 10/25/15.  You may need to refine/expand your conjecture first (let’s discuss this in class).   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 Tuesday, 11/6/15):  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 (created in class 10/25):

Group 1: Federico, Samantha, Yvan

Group 2: Jess, Young,  Rachel, Alex

Group 3:  Danping, Franklin,  Silvana, Jessie

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, 10/18/17). 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?

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 conjectures about Tic-Tac-Toe:
Conjecture: The person who goes first always wins.
Conjecture: The center is the best move.
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.”

The assignment below is due BEFORE CLASS on Thursday, October 11th (it is essential that you complete it before class, as we will be doing a class activity building on the assignment).

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 – both shores of a river and two islands in midstream – connected 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.

### Background – Graph Theory

We can simplify the picture of King’s Mountain to make it easier to deal with:

The key elements of the map are the four land masses (let’s label them A, B, C, and D) and the seven bridges (p,q,r,s,t,u and v) (thanks to mathisfun.com for the images):

For the purposes of our problem, we can simply think about each land mass as a point (A, B, C, and D), and the bridges as lines connecting the points (p,q,r,s,t,u and v) – like this:

We call this kind of picture a graph – the points are called vertices and the the lines are called edges.  Our goal of finding “a walking tour that crosses each bridge once” is now matter of tracing out all the edges without lifting our pencil (and without repeating any edge).

### Assignment, Due Thursday 10/11 (beginning of class)

Warm up (This Warm Up is just for practice – you do NOT need to submit your answers – see below for the three-part Assignment to be submitted).  The following examples build on the example above.

EXAMPLE 2: If you are given the freedom to build one new bridge in King’s Mountain (“make one new edge in the graph”), can you do it in such a way the walking tour becomes possible?  Do it!

EXAMPLE 3: If you are given the freedom to destroy one bridge (“erase one edge”), can you do it in such a way that the walking tour becomes possible? Do it!

EXAMPLE 4: Construct walking tours for each of the following graphs (or decide if it is impossible).

Assignment.  Your assignment has 4 parts.

PART 1.  For each of the four graphs below (G1 – G4), decide whether it is possible to create a walking tour crossing each bridge exactly once.  Post your solutions here (TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER).  If it is not possible to create a create a solution, say so!

PART 2.  Challenge your friends:  Now it’s up to you to build your own graph, and challenge your classmates to construct a walking tour (or to determine if it is impossible).  It can consist of as many points as you wish, and as many bridges (edges) connecting them.  You MUST label your points “A, B, C…” etc.  When you’re finished, decide for yourself if a walking tour crossing each bridge exactly once is possible.   Remember, the most challenging puzzles are the ones where the answer is difficult to determine. Post two puzzles in the comments.  See the note  “POSTING YOUR PUZZLE ONLINE” below for instructions on how to draw and share graphs online.

PART 3.  Solve a friend’s puzzle.  Leave a response to a friend’s posted puzzle, giving a solution.  TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER.

Example:
Here is a puzzle: http://sketchtoy.com/67467551
Here is a solution: (start at A) – A, B, D, A, E, B, C, E

PART 4.  The third part of your assignment is to write a short paragraph (at least 3 sentences) responding to the following prompt.  Be sure to respond to each part:

Writing Prompt:  Did you enjoy this assignment? Why or why not?  Describe a connection between this assignment and our work in the class.  (If you don’t believe there is a connection, try to imagine why we are doing this).  Leave your response in the comments.

POSTING YOUR PUZZLE ONLINE.  I recommend the site sketchtoy.com – it allows you to draw something, then click “SAVE” and get a link to your drawing.  You can post the link in a comment, and we’ll be able to click on it and view your drawing.   Don’t worry if it’s not pretty!  For example, here is a graph that I drew (can you find a walking tour that crosses all edges?): http://sketchtoy.com/67467556

Due Thursday, 9/20/18.  For this week’s writing assignment, take a look at the picture below called “Sentences.”  Read every sentence in the picture.  As you read, pay attention to your own stream of consciousness – what are the thoughts that pop into your head?  For full credit, respond to all 4 of the following items.

1. Record two observations about the sentences in the picture – what do you notice / what’s something you find interesting / what popped into your head / what stood out.
2. Choose one of the sentences in the picture (do NOT choose the same sentence as anyone else).  Type the sentence out “in quotes”, and then answer the following questions:
a. Is it a statement (as discussed in class on Tuesday)?  Explain why or why not.
b. Is it true or false? Explain in everyday English why or why not.
3. Make up a sentence that you believe would fit into this picture.  Tell us the sentence, and then tell us whether it is a statement, and whether it is true or false.
4. What connection (if any) does this assignment have to do with the work we are doing in class?

“Sentences” by Flickr user Eldeem

This assignment is due Thursday, September 13, at the start of class.

Assignment.  Choose ONE of the following two topics.  Write a reply to this post, responding to the topic.  Begin by telling us which topic you chose. (1-2 paragraphs).

Topics.

1. Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.
2. Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college).  Describe what happened.  Do you think you could explain it to others in a way that they could have the same flash of understanding?

Extra Credit.  For extra credit, write a response to one of your classmates’ comments.  Do you feel the same, or different?  Did you learn anything?  Did you get any ideas about teaching, or about learning?

Why are we doing this, anyway?  We are following two ideas that have come up already in class — things that may not seem related to learning math, but research shows that engaging in these activities can dramatically increase the amount that you learn, and change the way you learn it.  The first is writing – something not typically associated with mathematics.  When you express your ideas in words, it forces you to think them through very carefully, detail by detail.  A great way to check and see if you really understand something is to try to explain it to someone else, either out loud or in writing.  Example: if you know how to add fractions, try teaching it someone who doesn’t know how.  The second is called metacognition, or “thinking about thinking.”  This happens when you think about what was going on in your head while you were working on a problem or trying to learn a new idea.  What train of thought did you follow?  Where did you get stuck, and what did you do next?  What were you feeling at the time? and so on.  Combining writing and metacognition can be a tremendously powerful tool in identifying the ways we learn best and the ways we make mistakes, and learning to improve.  However, like any skill, it takes practice.  That’s why we’re getting started by writing a little about our past experiences with mathematics.

For the past several years I have taught this same course in the Fall semester.   At the end of each course, I give my students the following assignment:

Imagine that you are invited to speak on the first day of MAT 2071, to give advice to entering students.  Write at least three sentences … describing what you would tell them.

To see the assignment and the students’ responses, follow this link for Fall 2017 and  this link for Fall 2016.

Your assignment, due at the beginning of class on Thursday, September 6th, is to:

1. Read through ALL the responses (there are 25 of them altogether).
2. Write a reply to this post (1 paragraph) responding to all of the following:
1. What advice seemed most relevant to you personally? Why? (you can copy/paste a short statement, or put it in your own words)
2. Based on this advice, what changes can you make right now to help you succeed in this course?

Extra Credit. For extra credit, write a response to one of your classmates’ comments.  Do you have any advice?  Be kind.

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