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what is mathematical inquiry? 2017 IBL Workshop @ Cal Poly Dana C. Ernst June 29, 2017

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deep practice

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deep practice Take 45 seconds to look over the following list of pairs of words, but do not write anything down. bread/b tter ocean/breeze leaf/tree music/l rics sweet/sour sh e/sock phone/bo k movie/actress chi s/salsa gasoline/engine high school/college pen il/paper river/b at turkey/stuffing fruit/vegetable be r/wine computer/chip television/rad o l nch/dinner chair/couch 2

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deep practice Directions ∙ Without looking at the list of pairs of words, write down as many pairs as you can. You do not need to remember where any missing letters were nor which column/what order a pair was in. 3

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deep practice Directions ∙ Without looking at the list of pairs of words, write down as many pairs as you can. You do not need to remember where any missing letters were nor which column/what order a pair was in. ∙ Looking at the table on the next slide count how many pairs are in column A versus column B. 3

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deep practice A B ocean/breeze bread/b tter leaf/tree music/l rics sweet/sour sh e/sock movie/actress phone/bo k gasoline/engine chi s/salsa high school/college pen il/paper turkey/stuffing river/b at fruit/vegetable be r/wine computer/chip television/rad o chair/couch l nch/dinner Table: Word list from The Talent Code. 4

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deep practice According to The Talent Code by Daniel Coyle, studies show that on average people remember 3 times as many pairs in column B, the one with missing letters. Maybe a room full of mathematicians will have wildly different results, but … 5

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deep practice According to The Talent Code by Daniel Coyle, studies show that on average people remember 3 times as many pairs in column B, the one with missing letters. Maybe a room full of mathematicians will have wildly different results, but … You are peculiar! You are peculiar! You are peculiar! 5

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deep practice According to The Talent Code by Daniel Coyle, studies show that on average people remember 3 times as many pairs in column B, the one with missing letters. Maybe a room full of mathematicians will have wildly different results, but … You are peculiar! You are peculiar! The claim is that a microsecond of struggle (cognitive demand) makes all the difference. You are peculiar! 5

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deep practice According to The Talent Code by Daniel Coyle, studies show that on average people remember 3 times as many pairs in column B, the one with missing letters. Maybe a room full of mathematicians will have wildly different results, but … You are peculiar! You are peculiar! The claim is that a microsecond of struggle (cognitive demand) makes all the difference. Ponder: What does this have to do with teaching? You are peculiar! 5

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how logical are our students?

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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: 7

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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. 7

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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? 7

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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? 8

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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. 9

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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. 9

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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. 9

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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. ∙ This activity underscores why we need mathematical systems to support our thinking. 9

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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. ∙ This activity underscores why we need mathematical systems to support our thinking. 9

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stepping stones 10

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what is mathematical inquiry?

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inquiry framework Part 1 Explore mathematical ideas related to Lights On. As you explore, keep a record of the process: Explore Lights On ∙ Where is the mathematics? ∙ Record all mathematical ideas and questions. Meta-Process ∙ Record any mathematician moves you make. ∙ Keep track of the motivation behind those moves. Our objective is to become consciously aware of the questions we ask and moves we make while doing mathematics. 12

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inquiry framework Part 1 Explore mathematical ideas related to Lights On. As you explore, keep a record of the process: Explore Lights On ∙ Where is the mathematics? ∙ Record all mathematical ideas and questions. Meta-Process ∙ Record any mathematician moves you make. ∙ Keep track of the motivation behind those moves. Ponder: How do your moves look in other mathematical contexts? 15

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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. 16

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inquiry framework Here is a non-mathematical example. 17

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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, and ∙ The behaviors you’ve seen in your students. The goal is to produce a visual representation for the process of mathematical inquiry. 18