Sum 2012 Day 1 Presentation, Beyond Pólya: Making Mathematical Habits of Mind an Integral Part of the Classroom
1. 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
2. Goals
Brainstorm mathematical habits of mind
Brainstorm ways to teach these habits
Explore strategies I use
Do some math problems
Reflect on this experience
3. Math as a noun We Fall Short
Where
Content
Few available resources
Easier to implement with good problems, but good
Math as are verb find/create.
problems a hard to
Not how I learned the subject
Mathematical Habits of Mind
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?
4. Mathematical Habits of Mind: My
version
1. Stop, Collaborate and Listen
A. Actively listens and engages
B. Asks for clarification when necessary
C. Challenges others in a respectful way
when there is disagreement
D. Promotes equitable participation
E. Willing to help others when needed
F. Believes the whole is greater than the sum
of its parts
G. Gives others the opportunity to have “aha”
moments
5. Mathematical Habits of Mind:
1. Stop, Collaborate and Listen
A. Actively listens and engages
B. Asks for clarification when necessary
Once upon C. Challenges others in a antidote to a poison was a stronger poison,
a time there was a land where the only respectful way
which needed to be the next drink after the first poison. In this land, a malevolent dragon
challenges the country’s wise king is a duel. The king has no choice but to accept.
when there to disagreement
The rules ofD. duel are such: Each dueler brings a full cup. First they must drink half of their
the Promotes equitable participation
opponent’s cup and then they must drink half of their own cup.
E. Willing to help others when needed
The dragonF.able to fly toothers the opportunitypoison in the “aha” is located.
is Gives a volcano, where the strongest to have country
The king doesn’t have the dragon’s abilities, so there is no way he can get the strongest
moments
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
6. 2. Persevere and Reflect
A. Can begin a problem independently
B. Works on one problem for greater and greater lengths of
time
C. Spends more and more time stuck without giving up
D. Can reduce or eliminate "solution path tunnel vision"
E. Contextualizes problems
F. Determines if answer is reasonable through analysis
G. Determines if there are additional or easier explanations
H. Embraces productive failure
7. 3. Describe
A. Conversational, verbal, and written articulation of
thoughts, results, conjectures, arguments, process,
proofs, questions, opinions
B. Can explain both how and why
C. Invents notation and language when helpful
D. Creates precise problems and notation
8. 4. Experiment and Invent
A. Creates variations
B. Creates generalizations
C. Creates extensions
D. Looks at simpler examples when necessary
E. Looks at more complicated examples when
necessary/interesting
F. Creates and alters rules of a game
G. Invents new mathematical systems that are innovative,
but not arbitrary
9. 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
they 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:
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
10. 5. Pattern Sniff
A. On the lookout for patterns
B. 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:
1. You can only use fractions with 1 in the numerator
2. You are allowed to write fractions as addition problems (you can add
more than two fractions together)
3. Every fraction in your addition problem must be different
11. 6. Guess and Conjecture
A. Guesses
B. Estimates
C. Conjectures
D. Healthy skepticism of experimental results
E. Determines lower and upper bounds
F. Looks at special cases to find and test conjectures
G. Works backwards (guesses at a solution and see if it
makes sense)
12. 7. Strategize, Reason, and Prove
A. Moves from data driven conjectures to theory based
conjectures
B. Searches for counter-examples
C. Proves conjectures using reasoning
D. Identifies mistakes or holes in proposed proofs by
others
E. Uses different proof techniques (inductive, indirect,
etc)
F. Strategizes about games such as “looking ahead”
13. 7. Strategize, Reason, and Prove
A. Moves from data driven conjectures to theory based
conjectures
B. SearchesThe Game of 21 Nim
for counter-examples
C.
RulesProves conjectures using reasoning
1. D.player game
2 Identifies mistakes or holes in proposed proofs by
2. Start with 21 “stones”
others
3. In each turn, a player removes 1, 2, or 3 stones. You
E. Uses different proof techniques (inductive, indirect,
must remove at least 1 stone.
etc)
4. The player who removes the last stone wins.
F. Strategizes about games such as “looking ahead”
14. 8. Organize and Simplify
A. Records results in a useful and flexible way (t-table,
state, Venn & tree diagrams)
B. Considers different forms of answers
C. ? ?
Process, solutions and answers are organized and
easy to follow
D. Determine whether the problem can be broken up
into simpler pieces
E. Uses methods to limit and classify cases (parity,
partitioning)
F. Uses units of measurement to develop and check
formulas
15. 8. Organize and Simplify
A. Records results in a useful and flexible way (t-table,
state, Venn & tree diagrams)
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
16. 9. Visualize
A. Uses pictures/placement to describe and solve problems
B. Uses manipulatives to describe and solve problems
C. Reasons about shapes
D. Visualizes data
E. Looks for symmetry
F. Visualizes relationships (using tools such as Venn
diagrams and graphs)
G. Visualizes processes (using tools such as graphic
organizers)
H. Visualizes changes
I. Visualizes calculations (such as mental arithmetic)
17. 10. Connect
A. Articulates how different skills and concepts are
related
B. Applies old skills and concepts to new material
C. Describes problems and solutions using multiple
representations
D. Finds and exploits similarities within and between
problems
18. 10. Connect
A. Articulates how different skills and concepts are
related
B. Applies old skills and of 15 Cats new material
Game concepts to
C.
RulesDescribes problems and solutions using multiple
representations
1. 2 player game
2. D. Finds alternate picking a number between 1 and 9
Players and exploits similarities within and between
problems
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.
19. 10. Connect
15 Cats
Rules
1. 2 player game Magic Squares
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.
20. Avery Pickford
The Nueva School
Mills College
Blog: Without Geometry, Life is Pointless @
www.withoutgeometry.com
@woutgeo
avery@withoutgeometry.com
Editor's Notes
Re not sure they are valued: Really not sure they are valued beyond their ability to help students access content.
Brainstorm how people promote collaboration and listening in their classroomStart with this as I’d also like this to be a framework for todayB. 10 second pauseC. Try and create situations where this happens by using open ended questions with multiple solutions and/or multiple solution methodsD. If no one has questions, I ask them to formulate questions someone else might have if they were confused: metacognitive work about what people might find challengingE. HardF. Group answer of the population of Istanbul: 13.5 million. When finance professor Jack Treynor ran the experiment in his class with a jar that held 850 beans, the group estimate was 871. Only one of the fifty-six people in the class made a better guess. Wisdom of Crowds by James SurowieckiG. Encourage appropriate hints instead of telling answers
Brainstorm how people promote collaboration and listening in their classroom (brainstorm at tables and have each table share out something someone else said)Start with this as I’d also like this to be a framework for todayC. Try and create situations where this happens by using open ended questions with multiple solutions and/or multiple solution methodsD. HardE. HardF. Encourage appropriate hints instead of telling answers
Brainstorm: What does this mean and how do we do it? (share at tables and have people share out one thing someone else said)A. Made easier by problems that limit terminology/symbolsB. Death to 1-29 oddC. Need to give students time to work in open ended, low stakes environmentsD. Big one for me. Too many “strong” math students who don’t experience failure enough. Too many “weak” math students whose failures are not productive or appreciatedD. Alan Schoenfeld at UC Berkeley found that one of the most telling differences between novice and expert mathematicians were the ability of the latter group to abandon unhelpful pathsE. What’s important and what’s not (what are the rules for this problem)?F. Does my answer make senseG. To reiterate the importance of failure, count off…1, 2, ?(example of productive failure AND our predilection for pattern sniffing)
Thinking about the progression from conversation to verbal to writtenB. Importance of conceptual understandingD. One reason to not make my problems precise
Variations are small tweaks to a problem, such as changing the numbersGeneralizations explore sets of problemsEvery problem has explicit and implicit constraints and rules. Extensions alter these constraints and/or rules to create a new problem inspired by the first.
Clarifying questions about rules.Explore, create some problems, create some extensions, variations, and generalizationsShare out
Should be at 1 hour markGuess -> Evidence -> Conjecture -> Proof cycle with a healthy skepticism throughoutA. Write down answers that are too high/too low. Encouraging guessing also destigmatizes being wrong.E. Determining lower and upper bounds: “Say an answer you know is too high.” A way to scaffold and feel intermediate success.*Problem Solving Salute: Working backwards
A. T-tables are our best friend, but we sometime gloss over the most important aspect: what data do we collect?B. Sometimes 6/36 is “simpler” than 1/6D. No reason we can’t start using vocabulary like lemmas in elementary school
Play Penney’s Game
The geometric representation of imaginary #’sI was teaching before I connected “completing the square” with an actual square.D. Lots of cool new ways to visualize statistics (Hans Rosling)E. Looking at the state diagrams for Penney’s Game
Everything will be available for public ridicule shortly.
