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

motivation

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

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

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

game plan

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

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

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

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

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 10

example problems

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

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

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? 14

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

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. 16

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

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? 18

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

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? 20

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? 21

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. 22

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? 23