2019 IBL Workshop What is Mathematical Inquiry? Portland, OR June 25–28, 2019

What is Mathematical Inquiry? 0

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. 1

Inquiry Framework 2

Inquiry Framework 3

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? 4

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. 5

A Non-Mathematical Example http://www.sciencebuddies.org 6

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. 7

How logical are our students? 7

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 op- posite face is blue. Which cards must you turn over in order to test the truth of this statement without turning over any unnecessary cards? 8

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? 9

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. 10