Transitioning students from consumers to producers

In response to assessment reports identifying weaknesses in communication and reasoning of junior and senior mathematics majors, we have developed a 3-credit semester-long course that is required for all first-year mathematics majors. The focus of this course is on reasoning and communication through problem solving and written mathematical arguments in order to provide students with more experience and training early in their university studies. The goal is for the students to work on interesting yet challenging multi-step problems that require almost zero background knowledge. The hope is that students will develop (or at least move in the direction of) the habits of mind of a mathematician. The problem solving of the type in the course is a fundamental component of mathematics that receives little focused attention elsewhere in our program. The course will be taught via an inquiry-based learning (IBL) approach with an explicit focus on students asking questions and developing conjectures. In this talk, we will describe the structure of the course and our plan for transitioning students from “consumers” of mathematics to “producers”.

This talk was presented in the MAA Contributed Paper Session on Teaching Inquiry on January 12, 2015 at the 2015 Joint Mathematics Meetings in San Antonio, TX.

Transcript

1. transitioning students from consumers to
   producers
   MAA Session on Teaching Inquiry
   Dana C. Ernst & Nándor Sieben
   Northern Arizona University
   January 12, 2015
   

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2. motivation
   

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3. 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 Habit 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.”
   2
   

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4. motivation
   Claims
   1. An education must prepare students to ask & explore questions in
   contexts that do not yet exist.
   2. If we really want students to be independent, inquisitive, &
   persistent, then we need to provide them with the means to
   acquire these skills.
   3
   

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5. motivation
   Department assessment reports have identified weaknesses in
   communication & reasoning of junior & senior mathematics majors.
   My Lofty Goals
   1. Transition students from consumers to producers!
   2. I want to provide the opportunity for a transformative experience.
   3. I want to change my students’ lives!
   4
   

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6. game plan
   

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7. game plan
   Introduction to Mathematical Reasoning
   We have developed a 3-credit semester-long course that is required
   for all 1st-year math majors: Introduction to Mathematical
   Reasoning. Prerequisite: Calculus 1.
   The focus is on reasoning & communication through problem solving
   & written mathematical arguments in order to provide students with
   more experience & training early in their university studies.
   6
   

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8. game plan
   Introduction to Mathematical Reasoning (continued)
   Plan: Students work on interesting yet challenging multi-step
   problems that require almost zero background knowledge.
   Hope/Goal: Students move in the direction of the habits of mind of
   a mathematician.
   7
   

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9. game plan
   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.”
   8
   

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10. game plan
    Course Structure & Approach
    1. Inquiry-based learning (IBL) approach.
    2. Informal student presentations of progress on previously assigned
    homework problems.
    3. Exploration of alternative approaches, possible generalizations,
    consequences, special cases, converse.
    4. Discussion of relationships to previously assigned or solved
    problems.
    5. Group work focused on problems selected by the instructor.
    6. Explanation of unfamiliar mathematical concepts as needed.
    9
    

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11. game plan
    Types of Assignments
    1. Daily Homework: Chip away at problem sequence. Focus of
    student presentations.
    2. Weekly Homework: Formal write-ups, revise solutions to Daily
    Homework.
    3. Coding Homework: Utilize small computer programs (Python) to
    solve problems.
    Resources
    1. Our own imagination
    2. Problems from similar courses
    3. Hungarian high school problem sets
    4. Problems from Math Circles & Math Teacher Circles
    5. Project Euler
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12. example problems
    

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13. example problems
    Warning!
    I’m not going to tell you what any of the answers are!
    Problem
    An ant is crawling along the edges of a unit cube. What is the
    maximum distance it can cover starting from a corner so that it does
    not cover any edge twice?
    12
    

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14. example problems
    Problem
    A mouse eats his way through a 3 × 3 × 3 cube of cheese by
    tunneling through all of the 27 1 × 1 × 1 sub-cubes. If she starts at
    one corner & always moves to an uneaten sub cube, can she finish
    at the center of the cube?
    13
    

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15. example problems
    Problem
    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 two-pan scale.
    Problem
    Consider the situation in the previous problem, but suppose you
    have n coins. For which n is it possible to devise a procedure for
    identifying the counterfeit coin in only 3 weighings with a two-pan
    scale?
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16. example problems
    Problem
    The Sylver Coinage Game is a game in which 2 players alternately
    name positive integers that are not the sum of nonnegative
    multiples of previously named integers.
    The person who names 1 is the loser! If player 1 names 16, & both
    players play optimally thereafter, then who wins?
    15
    

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17. example problems
    Problem (continued)
    A sample game between A & B
    ∙ A opens with 5. Now neither player can name 5, 10, 15, . . .
    ∙ B names 4. Now neither player can name 4, 5, 8, 9, 10, or any
    number greater than 11.
    ∙ A names 11. Now the only remaining numbers are 1, 2, 3, 6, & 7.
    ∙ B names 6. Now the only remaining numbers are 1, 2, 3, & 7.
    ∙ A names 7. Now the only remaining numbers are 1, 2, & 3.
    ∙ B names 2. Now the only remaining numbers are 1 & 3.
    ∙ A names 3, leaving only 1.
    ∙ B is forced to name 1 & loses.
    This is an unsolved Conway problem with a monetary reward.
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18. example problems
    Problem
    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?
    17
    

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19. example problems
    Problem
    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?
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20. example problems
    Problem
    There are n very intelligent lions on an inhabited island. They are
    very hungry because they ate everything they could. So whenever
    there is a new food source, the closest lion eats all the available
    food & falls asleep to digest. A sleeping lion becomes prey. What
    happens if a helicopter drops a dead gazelle onto the island?
    19
    

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21. example problems
    Problem
    Imagine you have 25 pebbles, each occupying one square on a 5 × 5
    chess board. Tackle each of the following variations of a puzzle.
    1. Variation 1: 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.
    2. Variation 2: Suppose that all but one pebble (your choice which
    one) 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.
    3. Variation 3: Consider Variation 1 again, but this time also allow
    diagonal moves to adjacent squares. If this is possible, describe a
    solution. If this is impossible, explain why.
    4. Variation 4: What about other size boards?
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22. example problems
    Problem
    The first vote counts of the papal conclave resulted in 33 votes each
    for candidates A & B and 34 votes for candidate C. The cardinals
    then discussed the candidates in pairs. In the second round each
    pair of cardinals with differing first votes changed their votes to the
    third candidate they did not vote for in the first round. The new vote
    counts were 16, 17 & 67. They were about to start the smoke signal
    when Cardinal Ordinal shouted “wait”. What was his reason?
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23. example problems
    Problem
    In a PE class, everyone has 5 friends. Friendships are mutual. Two
    students in the class are appointed captains. The captains take
    turns selecting members for their teams, until everyone is selected.
    Prove that at the end of the selection process there are the same
    number of friendships within each team.
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24. example problems
    Problem
    Suppose you have 6 toothpicks that are exactly the same length. Can
    you arrange the toothpicks so that 4 identical triangles are formed?
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