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.
Sum 2012 Day 1 Presentation, Beyond Pólya: Making Mathematical Habits of Mind an Integral Part of the Classroom
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: email@example.com Backchannel: http://todaysmeet.com/SUM
Goals Brainstorm mathematical habits of mind Brainstorm ways to teach these habits Explore strategies I use Do some math problems Reflect on this experience
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?
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
Mathematical Habits of Mind: 1. Stop, Collaborate and Listen A. Actively listens and engages B. Asks for clarification when necessaryOnce upon C. Challenges others in a antidote to a poison was a stronger poison, a time there was a land where the only respectful waywhich needed to be the next drink after the first poison. In this land, a malevolent dragonchallenges the country’s wise king is a duel. The king has no choice but to accept. when there to disagreementThe rules ofD. duel are such: Each dueler brings a full cup. First they must drink half of their the Promotes equitable participationopponent’s cup and then they must drink half of their own cup. E. Willing to help others when neededThe dragonF.able to fly toothers the opportunitypoison in the “aha” is located. is Gives a volcano, where the strongest to have countryThe king doesn’t have the dragon’s abilities, so there is no way he can get the strongest momentspoison. The dragon is confident of winning because he will bring the stronger poison. How canthe king kill the dragon and survive? Adapted from Tanya Khovanova’s Math Blog http://blog.tanyakhovanova.com
2. Persevere and ReflectA. Can begin a problem independentlyB. Works on one problem for greater and greater lengths of timeC. Spends more and more time stuck without giving upD. Can reduce or eliminate "solution path tunnel vision"E. Contextualizes problemsF. Determines if answer is reasonable through analysisG. Determines if there are additional or easier explanationsH. Embraces productive failure
3. DescribeA. Conversational, verbal, and written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinionsB. Can explain both how and whyC. Invents notation and language when helpfulD. Creates precise problems and notation
4. Experiment and InventA. Creates variationsB. Creates generalizationsC. Creates extensionsD. Looks at simpler examples when necessaryE. Looks at more complicated examples when necessary/interestingF. Creates and alters rules of a gameG. Invents new mathematical systems that are innovative, but not arbitrary
4. Experiment and InventSome of Egyptian mathematics looked quite different from the math we usetoday. For example, Egyptians had no way to write a fraction with anythingbut 1 in the numerator. So no 3/5 or 5/7 or 13/10. If they wanted to describethey just wrote this as a sum of distinct unit fractions. So instead of writing5/8, they would write 1/2+1/8.So to recap the rules:1. Egyptians only use fractions with 1 in the numerator2. 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
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 numerator2. 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
6. Guess and ConjectureA. GuessesB. EstimatesC. ConjecturesD. Healthy skepticism of experimental resultsE. Determines lower and upper boundsF. Looks at special cases to find and test conjecturesG. Works backwards (guesses at a solution and see if it makes sense)
7. Strategize, Reason, and ProveA. Moves from data driven conjectures to theory based conjecturesB. Searches for counter-examplesC. Proves conjectures using reasoningD. Identifies mistakes or holes in proposed proofs by othersE. Uses different proof techniques (inductive, indirect, etc)F. Strategizes about games such as “looking ahead”
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 reasoning1. D.player game 2 Identifies mistakes or holes in proposed proofs by2. Start with 21 “stones” others3. 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”
8. Organize and SimplifyA. Records results in a useful and flexible way (t-table, state, Venn & tree diagrams)B. Considers different forms of answersC. ? ? Process, solutions and answers are organized and easy to followD. Determine whether the problem can be broken up into simpler piecesE. Uses methods to limit and classify cases (parity, partitioning)F. Uses units of measurement to develop and check formulas
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)Rules1. 2 player game2. 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 recorded4. The player whose sequence appears first wins
9. VisualizeA. Uses pictures/placement to describe and solve problemsB. Uses manipulatives to describe and solve problemsC. Reasons about shapesD. Visualizes dataE. Looks for symmetryF. Visualizes relationships (using tools such as Venn diagrams and graphs)G. Visualizes processes (using tools such as graphic organizers)H. Visualizes changesI. Visualizes calculations (such as mental arithmetic)
10. ConnectA. Articulates how different skills and concepts are relatedB. Applies old skills and concepts to new materialC. Describes problems and solutions using multiple representationsD. Finds and exploits similarities within and between problems
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 representations1. 2 player game2. 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, its a tie.
10. Connect 15 CatsRules1. 2 player game Magic Squares2. 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, its a tie.
Avery Pickford The Nueva School Mills CollegeBlog: Without Geometry, Life is Pointless @ www.withoutgeometry.com @woutgeo firstname.lastname@example.org