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IBL Workshop DePaul University June 19–22, 2018

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How logical are our students? 0

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How logical are our students? Here are four cards lying flat on a table. Each card has a single-digit number on one side and one of two colors (blue or green) on the other side. Consider the following statement: 1

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How logical are our students? Here are four cards lying flat on a table. Each card has a single-digit number on one side and one of two colors (blue or green) on the other side. Consider the following statement: If a card shows an even number on one face, then its opposite face is blue. 1

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How logical are our students? Here are four cards lying flat on a table. Each card has a single-digit number on one side and one of two colors (blue or green) on the other side. Consider the following statement: If a card shows an even number on one face, then its opposite face is blue. Which cards must you turn over in order to test the truth of this statement without turning over any unnecessary cards? 1

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How logical are our students? Imagine yourself as a police officer in a bar looking for underage drinkers. The rule is: If a person is drinking beer, then that person must be over 21. You see four people: Which people do you need to check to make sure the rule is being followed? 2

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How logical are our students? • When presented the number/color task in psychology experiments, only 10% of people selected the right answer. 3

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How logical are our students? • When presented the number/color task in psychology experiments, only 10% of people selected the right answer. • When the task was reframed in the underage drinking context, 75% of people got the right answer. 3

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How logical are our students? • When presented the number/color task in psychology experiments, only 10% of people selected the right answer. • When the task was reframed in the underage drinking context, 75% of people got the right answer. • Psychologists: When given abstract tasks, the brain cuts corners and we act irrationally. 3

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How logical are our students? • When presented the number/color task in psychology experiments, only 10% of people selected the right answer. • When the task was reframed in the underage drinking context, 75% of people got the right answer. • Psychologists: When given abstract tasks, the brain cuts corners and we act irrationally. • Underscores why we need mathematical frameworks to support our thinking. 3

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What is Mathematical Inquiry? 3

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Inquiry Framework Part 1 Explore mathematical ideas related to Blink. As you explore, keep a record of the process: Explore Blink • Where is the mathematics? • Rules, conjectures, proofs, generalizations? • Record all mathematical ideas and questions. Meta-Process • Record any mathematical moves you or your peers make. • If possible: What is the motivation behind those moves? Our objective is to become consciously aware of the questions we ask and moves we make while doing mathematics. 4

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Inquiry Framework Part 1 Explore mathematical ideas related to Blink. As you explore, keep a record of the process: Explore Blink • Where is the mathematics? • Rules, conjectures, proofs, generalizations? • Record all mathematical ideas and questions. Meta-Process • Record any mathematical moves you or your peers make. • If possible: What is the motivation behind those moves? Ponder: How do your moves look in other mathematical contexts? 7

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Inquiry Framework Part 2 Now that we have the raw data of questions and moves, our objective is to organize our thinking into a visual representation of the process of mathematical inquiry. 8

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A Non-Mathematical Example http://www.sciencebuddies.org 9

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Inquiry Framework Part 2 Consider the following sources of information about the process of doing mathematics: • The ideas generated in Part 1 • Your own experiences doing mathematics for fun/research • The behaviors you’ve seen in your students The goal is to produce a visual representation for the process of mathematical inquiry. 10