Fancy Sounding Title

Beyond Pólya:

Making Mathematical Habits of Mind an Integral Part of the Classroom Part Deux

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

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Goals

• Recap mathematical habits of mind
• Do some math problems
• Create and share habits of mind lessons
• Create and share habits of mind assessments
• Reflect on this experience

Pick a problem and work on it…

1) Determine the subgroup lattice of GL(2; 2).��2) What is the minimum number of guests that must be invited to a party so that at least 5 people will know each other or at least 5 people will not know each other?

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For our students:

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Starts with good problems

• Accessible
• Minimizes vocabulary and notation
• Is only as precise as necessary
• Has multiple entry points
• Includes ways to collect data
• Has multiple solution methods
• Deep
• Naturally leads to variations, generalizations, and extensions
• Leads to and connects different aspects of mathematics
• Motivates developing procedures, vocabulary, notation, and mathematical concepts
• Can be worked on for as long as you want
• Captivating
• Consists of benchmarks along the way where one is re-energized by the feeling of success
• Could be real world, but doesn’t have to be (and certainly shouldn’t be pseudo-contextual)
• May lead to a surprising result
• May feel like a puzzle waiting to be solved
• May be necessary to solve a different, interesting problem (which is not the same as “you’ll need to know this next year”).
• May be posed by students.
• Sideways scalable.
• Assessable in meaningful ways
• Can be scaffolded
• Mathematical.
• Problem solving skills and/or the language of mathematics help make progress in defining, simplifying, quantifying, dividing and/or solving the problem. Exploring the problem promotes mathematical habits of mind. 

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But enough about me…�Time to do some math

Draw a rectangle on a square grid. An example 9 by 3 rectangle is drawn

for you below. Draw one diagonal. How many squares does the diagonal

pass through?

Develop a rule to determine the number of squares a diagonal passes

through for any rectangle of any size.

Tasks

*Do some math

*Think about where students might get stuck & helpful habits that could get them unstuck

Organize & Simplify: Student work

Organize & Simplify (less successful)

Organize and Simplify

Tinker and Invent

3D Version

Mathematical Habits of Mind: My version

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.

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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.

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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?

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Adapted from Tanya Khovanova’s Math Blog

http://blog.tanyakhovanova.com

2. Persevere and Reflect

1. Collaborate and Listen

3. Describe

D. Creates precise problems and notation

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

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they just wrote this as a sum of distinct unit fractions. So instead of writing 5/8, they would write 1/2+1/8.

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So to recap the rules:

1. Egyptians only use fractions with 1 in the numerator
2. Egyptians write non-unit fractions as addition problems (you can add three or more fractions together)
3. Every fraction in an addition problem must be different

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A.Creates variations

B.Creates generalizations

C.Creates extensions

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5. Pattern Sniff

A. On the lookout for patterns

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6. Guess and Conjecture

D. Healthy skepticism of experimental results

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7. Strategize, Reason, and Prove

The Game of 21 Nim

Rules

1. 2 player game
2. Start with 21 “stones”
3. In each turn, a player removes 1, 2, or 3 stones. You must remove at least 1 stone.
4. The player who removes the last stone wins.

• Strategizes about games such as “looking ahead”

A. Records results in a useful and flexible way (t-table, state, Venn & tree diagrams)

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8. Organize and Simplify

The Penny Game (Penney’s Game)

Rules

1. 2 player game
2. Each player starts with a different sequence of three heads and tails (such as HHT vs. HTH)
3. One coin is flipped and the results recorded
4. The player whose sequence appears first wins

9. Visualize

E. Looks for symmetry

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D. Finds and exploits similarities within and between problems

10. Connect

Game of 15 Cats

Rules

1. 2 player game
2. 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.
3. 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.

Your turn…

Avery Pickford

The Nueva School

Mills College

Blog: Without Geometry, Life is Pointless @ www.withoutgeometry.com

@woutgeo

avery@withoutgeometry.com

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