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Jamming the Death Star cafeteria pool table with counterfeit coins: The wonderful world of MAT 220

Dana Ernst
January 31, 2017

Jamming the Death Star cafeteria pool table with counterfeit coins: The wonderful world of MAT 220

Dana, Monika, and Nandor will talk about some material from the problem solving course MAT 220.

This talk was given on January 31, 2017 in the Northern Arizona University Department of Mathematics and Statistics Colloquium.

Dana Ernst

January 31, 2017
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  1. jamming the death star cafeteria pool table
    with counterfeit coins
    The wonderful world of MAT 220
    Dana C. Ernst, Monika Keindl, & Nándor Sieben
    Northern Arizona University
    January 24, 2017

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  2. what is mat 220?
    Big Picture
    ∙ Focus of course is on reasoning & communication through
    problem solving.
    ∙ Goal is for students to work on interesting yet challenging
    multi-step problems that require little background knowledge.
    ∙ Hope is that students will develop the habits of mind of a
    mathematician.
    ∙ The problem solving of the type in this course is a fundamental
    component of mathematics that receives little focused attention
    elsewhere in our program.
    1

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  3. what is mat 220?
    A Hungarian-like Approach
    “In the Hungarian approach to learning and teaching, a strong and ex-
    plicit emphasis is placed on problem solving, mathematical creativity,
    and communication. Students learn concepts by working on problems
    with complexity and structure that promote perseverance and deep
    reflection.”
    “In the Hungarian approach to learning and teaching, a strong and ex-
    plicit emphasis is placed on problem solving, mathematical creativity,
    and communication. Students learn concepts by working on problems
    with complexity and structure that promote perseverance and deep
    reflection.”
    Ryota Matsuura (St. Olaf College & North American Director of
    Budapest Semesters in Mathematics Education) in On Teaching &
    Learning in Mathematics blog post.
    “In the Hungarian approach to learning and teaching, a strong and ex-
    plicit emphasis is placed on problem solving, mathematical creativity,
    and communication. Students learn concepts by working on problems
    with complexity and structure that promote perseverance and deep
    reflection.”
    2

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  4. what is mat 220?
    Course Structure
    ∙ Inquiry-based learning (IBL) approach.
    ∙ Informal student presentations of progress on previously assigned
    homework problems.
    ∙ Exploration of alternative approaches, possible generalizations,
    consequences, special cases, converse.
    ∙ Discussion of relationships to previously assigned or solved
    problems.
    ∙ Group work focused on problems selected by the instructor.
    ∙ Explanation of unfamiliar mathematical concepts as needed.
    3

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  5. motivation
    Problem
    Three strangers meet at a taxi stand & decide to share a cab to cut
    down the cost. Each has a different destination but all are heading
    in more-or-less the same direction. Bob is traveling 10 miles, Sally is
    traveling 20 miles, & Mike is traveling 30 miles. If the taxi costs $2
    per mile, how much should each contribute to the total fare?
    4

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  6. motivation
    Problem
    Three strangers meet at a taxi stand & decide to share a cab to cut
    down the cost. Each has a different destination but all are heading
    in more-or-less the same direction. Bob is traveling 10 miles, Sally is
    traveling 20 miles, & Mike is traveling 30 miles. If the taxi costs $2
    per mile, how much should each contribute to the total fare?
    Remark
    In preparation for their book, The Five Elements of Effective Thinking,
    Burger & Starbird encountered “I don’t know” as the most common
    response.
    4

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  7. motivation
    Claims
    1. In addition to helping students develop procedural fluency &
    conceptual understanding, we must prepare them to ask &
    explore new questions after they leave our classrooms. Content is
    not enough.
    2. If we really want students to be independent, inquisitive, &
    persistent, then we need to provide them with the means to
    acquire these skills.
    5

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  8. motivation
    “Much more important than specific mathematical results are the
    habits of mind used by the people who create those results. […]
    Although it is necessary to infuse courses and curricula with modern
    content, what is even more important is to give students the tools
    they will need in order to use, understand, and even make mathe-
    matics that does not yet exist.”
    “Much more important than specific mathematical results are the
    habits of mind used by the people who create those results. […]
    Although it is necessary to infuse courses and curricula with modern
    content, what is even more important is to give students the tools
    they will need in order to use, understand, and even make mathe-
    matics that does not yet exist.”
    Cuoco, Goldenberg, & Mark in Habits of Mind: An Organizing
    Principle for Mathematics Curriculum.
    “Much more important than specific mathematical results are the
    habits of mind used by the people who create those results. […]
    Although it is necessary to infuse courses and curricula with modern
    content, what is even more important is to give students the tools
    they will need in order to use, understand, and even make mathe-
    matics that does not yet exist.”
    6

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  9. spring 2015 (first iteration)
    Demographics
    ∙ Students: 15 (6 females, 9 males)
    ∙ Majors
    ∙ Math BS: 4
    ∙ Math & Physics: 1
    ∙ Math BSEd: 2
    ∙ Engineering: 2
    ∙ CS: 2
    ∙ Theater: 1
    ∙ Exploratory: 3
    ∙ Very difficult students: 1
    7

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  10. spring 2015 (first iteration)
    Types of Assignments
    ∙ Daily Homework: Chip away at problem sequence. Focus of
    student presentations.
    ∙ Weekly Homework: Formal write-ups, revise solutions to Daily
    Homework. Typed using LaTeX.
    ∙ Coding Homework: Utilize small computer programs (Python) to
    solve problems.
    Exams
    ∙ 2 midterm exams and a final.
    ∙ Midterms consisted of an in-class part and a take-home part.
    ∙ Exams were a mixture of problems students had previously
    encountered, problems similar in nature to ones previously
    encountered, and new problems.
    8

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  11. spring 2015 (first iteration)
    Typical Day
    ∙ Class was divided up into small groups.
    ∙ Each group was tasked with discussing one the problems from the
    Daily Homework and attempting to come to consensus on a
    solution.
    ∙ After 10–20 minutes, groups took turns sharing out.
    ∙ My job was to moderate discussion and guide student thinking.
    ∙ Students encouraged to annotate their work with colored marker
    pens in light of class discussion.
    9

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  12. spring 2017 (current iteration)
    Demographics
    ∙ Students: 25 (5 females, 20 males)
    ∙ Majors
    ∙ Math BS: 10
    ∙ Math & Physics: 1
    ∙ Math & CS: 1
    ∙ Math BSEd: 0
    ∙ CS: 10
    ∙ Physics & Astro: 1
    ∙ Business Econ: 1
    ∙ Biology: 1
    10

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  13. spring 2017 (current iteration)
    Updates
    ∙ Ditched Weekly and Coding assignments.
    ∙ Homework (formerly Daily Homework) is more or less the same as
    before. Chip away at problem sequence.
    ∙ Day-to-day format roughly the same. Need to adjust for increased
    class size.
    ∙ Replaced 2 midterms exams with 7 “quizzes”; occur every 2 weeks.
    ∙ Each quiz consists of problems from the previous 2 weeks and
    some new problems.
    ∙ Final exam will likely have a take-home component.
    11

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  14. example problems
    Problem 1
    Imagine a hallway with 1000 doors numbered consecutively 1
    through 1000. Suppose all of the doors are closed to start with. Then
    some dude with nothing better to do walks down the hallway and
    opens all of the doors. Because the dude is still bored, he decides to
    close every other door starting with door number 2. Then he walks
    down the hall and changes (i.e., if open, he closes it; if closed, he
    opens it) every third door starting with door 3. Then he walks down
    the hall and changes every fourth door starting with door 4. He
    continues this way, making a total of 1000 passes down the hallway,
    so that on the 1000th pass, he changes door 1000. At the end of this
    process, which doors are open and which doors are closed?
    12

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  15. example problems
    Problem 2
    Imagine you have 25 pebbles, each occupying one square on a 5 × 5
    chess board. Suppose that each pebble must move to an adjacent
    square by only moving up, down, left, or right. If this is possible,
    describe a solution. If this is impossible, explain why.
    13

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  16. example problems
    Problem 3
    An overfull prison has decided to terminate some prisoners. The
    jailer comes up with a game for selecting who gets terminated: 10
    prisoners are to be lined up all facing the same direction. On the
    back of each prisoner’s head, the jailer places either a black or a red
    dot. Each prisoner can only see the color of the dot for all of the
    prisoners in front of them and the prisoners do not know how many
    of each color there are. The jailer tells the prisoners that if a
    prisoner can guess the color of the dot on the back of their head,
    they will live, but if they guess incorrectly, they will be terminated.
    The jailer will call on them in order starting at the back of the line.
    Before lining up the prisoners and placing the dots, the jailer allows
    the prisoners 5 minutes to come up with a plan that will maximize
    their survival. What plan can the prisoners devise that will maximize
    the number of prisoners that survive?
    14

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  17. example problems
    Problem 4
    Suppose you have 12 coins, all identical in appearance & weight
    except for one that is either heavier or lighter than the other 11
    coins. Devise a procedure to identify the counterfeit coin in only 3
    weighings with a two-pan scale. What about n coins?
    15

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  18. example problems
    Problem 5
    I have 10 sticks in my bag. The length of each stick is an integer. No
    matter which 3 sticks I try to use, I cannot make a triangle out of
    those sticks. What is the minimum length of the longest stick?
    16

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  19. example problems
    Problem 6
    Three actors and their three agents want to cross a river in a boat
    that is capable of holding only two people at a time, under the
    constraint that no actor can be in the presence of another agent
    unless their own agent is also present, because each agent is
    worried their rivals will poach their client. How should they cross the
    river with the least amount of rowing?
    17

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  20. example problems
    Problem 7
    We have two strings of pyrotechnic fuse. The strings do not look
    homogeneous in thickness but both of them have a label saying 4
    minutes. So we can assume that it takes 4 minutes to burn through
    either of these fuses. How can we measure a one minute interval?
    18

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  21. example problems
    Problem 8
    Take 15 poker chips or coins, divide into any number of piles with
    any number of chips in each pile. Arrange piles in adjacent columns.
    Take the top chip off every column and make a new column to the
    left. Repeat forever. What happens? Make conjectures about what
    happens when we change the number of chips.
    19

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  22. example problems
    Problem 9
    Four red ants and two black ants are walking along the edge of a one
    meter stick. The four red ants, called Albert, Bart, Debbie, and Edith,
    are all walking from left to right, and the two black ants, Cindy and
    Fred, are walking from right to left. The ants always walk at exactly
    one centimeter per second. Whenever they bump into another ant,
    they immediately turn around and walk in the other direction. And
    whenever they get to the end of a stick, they fall off. Albert starts at
    the left hand end of the stick, while Bart starts 20.2 cm from the left,
    Debbie is at 38.7cm, Edith is at 64.9cm and Fred is at 81.8cm. Chuck’s
    position is not known—all we know is that he starts somewhere
    between Bart and Debbie. Which ant is the last to fall off the stick?
    And how long will it be before he or she does fall off?
    20

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