SUM 2012 Presentation Day 1

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Beyond Pólya: Making Mathematical Habits of Mind an Integral Part of the Classroom

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Fancy Sounding Title Beyond Pólya : Making Mathematical Habits of Mind an Integral Part of the Classroom Where You Can Stalk Me The Nueva School : San Francisco, CA Mills College: Oakland, CA Blog: Without Geometry, Life is Pointless www.withoutgeometry.com Twitter: @ woutgeo Email: avery@withoutgeometry.com Backchannel: http:// todaysmeet.com /SUM

Goals:

Goals Brainstorm mathematical habits of mind Brainstorm ways to teach these habits Explore strategies I use Do some math problems Reflect on this experience

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Mathematical Habits of Mind Content Math as a noun Math as a verb Where We Fall Short Few available resources Easier to implement with good problems, but good problems are hard to find/create. Not how I learned the subject Not sure habits are valued. Can take longer to see success/need to redefine success Will we care if habits are not explicitly assessed? If not, how do we assess?

Mathematical Habits of Mind: My version:

Mathematical Habits of Mind: My version Actively listens and engages Asks for clarification when necessary Challenges others in a respectful way when there is disagreement Promotes equitable participation Willing to help others when needed Believes the whole is greater than the sum of its parts Gives others the opportunity to have “aha” moments 1 . Stop, Collaborate and Listen

Mathematical Habits of Mind::

Mathematical Habits of Mind: Actively listens and engages Asks for clarification when necessary Challenges others in a respectful way when there is disagreement Promotes equitable participation Willing to help others when needed Gives others the opportunity to have “aha” moments Once upon a time there was a land where the only antidote to a poison was a stronger poison, which needed to be the next drink after the first poison. In this land, a malevolent dragon challenges the country’s wise king to a duel. The king has no choice but to accept. The rules of the duel are such: Each dueler brings a full cup. First they must drink half of their opponent’s cup and then they must drink half of their own cup. The dragon is able to fly to a volcano, where the strongest poison in the country is located. The king doesn’t have the dragon’s abilities, so there is no way he can get the strongest poison. The dragon is confident of winning because he will bring the stronger poison. How can the king kill the dragon and survive ? Adapted from Tanya Khovanova’s Math Blog http ://blog.tanyakhovanova.com 1 . Stop, Collaborate and Listen

2. Persevere and Reflect:

2. Persevere and Reflect Can begin a problem independently Works on one problem for greater and greater lengths of time Spends more and more time stuck without giving up Can reduce or eliminate "solution path tunnel vision" Contextualizes problems Determines if answer is reasonable through analysis Determines if there are additional or easier explanations Embraces productive failure

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Conversational , verbal, and written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions Can explain both how and why Invents notation and language when helpful Creates precise problems and notation 3. Describe

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Creates variations Creates generalizations Creates extensions Looks at simpler examples when necessary Looks at more complicated examples when necessary/ interesting Creates and alters rules of a game Invents new mathematical systems that are innovative, but not arbitrary 4. Experiment and Invent

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4. Experiment and Invent Some of Egyptian mathematics looked quite different from the math we use today. For example, Egyptians had no way to write a fraction with anything but 1 in the numerator. So no 3/5 or 5/7 or 13/10. If they wanted to describe t hey just wrote this as a sum of distinct unit fractions. So instead of writing 5/8, they would write 1/2+1/8. So to recap the rules: Egyptians only use fractions with 1 in the numerator Egyptians write non-unit fractions as addition problems (you can add three or more fractions together ) Every fraction in an addition problem must be different

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5. Pattern Sniff On the lookout for patterns Looking for and creating shortcuts/procedures Revisiting Egyptian Fractions Create an algorithm to convert a particular group of fractions into Egyptian Fractions. Remember the rules: You can only use fractions with 1 in the numerator You are allowed to write fractions as addition problems (you can add more than two fractions together) Every fraction in your addition problem must be different

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6. Guess and Conjecture Guesses Estimates Conjectures Healthy skepticism of experimental results Determines lower and upper bounds Looks at special cases to find and test conjectures Works backwards (guesses at a solution and see if it makes sense)

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7. Strategize, Reason, and Prove Moves from data driven conjectures to theory based conjectures Searches for counter-examples Proves conjectures using reasoning Identifies mistakes or holes in proposed proofs by others Uses different proof techniques (inductive, indirect, etc ) Strategizes about games such as “looking ahead”

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7. Strategize, Reason, and Prove Moves from data driven conjectures to theory based conjectures Searches for counter-examples Proves conjectures using reasoning Identifies mistakes or holes in proposed proofs by others Uses different proof techniques (inductive, indirect, etc ) Strategizes about games such as “looking ahead” The Game of 21 Nim Rules 2 player game Start with 21 “stones” In each turn, a player removes 1, 2, or 3 stones. You must remove at least 1 stone. The player who removes the last stone wins.

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Records results in a useful and flexible way (t-table, state, Venn & tree diagrams) Considers different forms of answers Process, solutions and answers are organized and easy to follow Determine whether the problem can be broken up into simpler pieces Uses methods to limit and classify cases (parity, partitioning) Uses units of measurement to develop and check formulas ? ? 8. Organize and Simplify

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Records results in a useful and flexible way (t-table, state, Venn & tree diagrams ) 8. Organize and Simplify The Penny Game (Penney’s Game) Rules 2 player game Each player starts with a different sequence of three heads and tails (such as HHT vs. HTH) One coin is flipped and the results recorded The player whose sequence appears first wins

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9. Visualize Uses pictures/placement to describe and solve problems Uses manipulatives to describe and solve problems Reasons about shapes Visualizes data Looks for symmetry Visualizes relationships (using tools such as Venn diagrams and graphs) Visualizes processes (using tools such as graphic organizers) Visualizes changes Visualizes calculations (such as mental arithmetic)

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10. Connect Articulates how different skills and concepts are related Applies old skills and concepts to new material Describes problems and solutions using multiple representations Finds and exploits similarities within and between problems

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10. Connect Articulates how different skills and concepts are related Applies old skills and concepts to new material Describes problems and solutions using multiple representations Finds and exploits similarities within and between problems Game of 15 Cats Rules 2 player game Players alternate picking a number between 1 and 9 and putting this number in their pile. Once a number has been picked, it can’t be chosen again. The first person that can make 15 by summing three of their numbers wins. If we go through all 9 numbers without any one of us being able to add up to 15, it's a tie.

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10. Connect 15 Cats Rules 2 player game Players alternate picking a number between 1 and 9 and putting this number in their pile. Once a number has been picked, it can’t be chosen again. The first person that can make 15 by summing three of their numbers wins. If we go through all 9 numbers without any one of us being able to add up to 15, it's a tie. Magic Squares

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Avery Pickford The Nueva School Mills College Blog: Without Geometry, Life is Pointless @ www.withoutgeometry.com @ woutgeo avery@withoutgeometry.com

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