Teaching Through Problem Solving

IMAPP 2007/2008

PS HeuristicsPS Heuristics

• Polya (1945)Polya (1945)

• Mason, Burton, & Stacey (1985)

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Polya (1945)Polya (1945)

• How to Solve ItHow to Solve It– Understand the problem

Devise a Plan– Devise a Plan

– Carry out the Plan

L k B k– Look Back

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Mason, Burton, & Stacey (1985)Mason, Burton, & Stacey (1985) 

• PhasesPhases– Entry

Attack– Attack

– Revise

P• Processes– Specializing

– Generalizing

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Polya versus Mason et alPolya versus Mason et al

Polya (1945) Mason et al (1985)Polya (1945)

PS Heuristic

Mason et al. (1985)

PS Heuristic (Phases, Processes, States, 

Understand the Problem

& Rubric)

Phases (below):Problem

Devise a Plan Entry

Carry out the Plan

Look Back

Attack

ReviewReview

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Questions and further discussionQuestions and further discussion

• Problem Solving brings together deepProblem Solving brings together deep procedural understanding and conceptual understanding – two sides of the coin calledunderstanding  two sides of the coin called mathematical understanding 

• Develops habits of mind that fosters thinking• Develops habits of mind that fosters thinking mathematically

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AN EXISTENCE PROOFAN EXISTENCE PROOF

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Existence ProofExistence Proof

• Making visible one way to Teach throughMaking visible one way to Teach through Problem Solving – TIMSS Video Studies: Japanese 3– TIMSS Video Studies: Japanese 3

• Problems

• Problem SolvingProblem Solving

• Teaching through Problem Solving

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TIMSS Video Studies as a Window into Teaching through Problem Solving (J3)through Problem Solving (J3)

• Reverse engineering the lesson• Reverse engineering the lesson• Noticing• Paying attention to taken for granted reified• Paying attention to taken‐for granted reified practices and unpacking them to develop professional habits of mindprofessional habits of mind

Seeing something once is better than hearingSeeing something once is better than hearing about it one hundred times.

– ConfuciusConfucius

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First, the Offering ProblemFirst, the Offering Problem

• It has been one month since Ichiro’s mother entered the hospital. He has decided to give a prayer with his small brother at a local temple every morning so that she will be well soon There are 18 ten yen coins inshe will be well soon. There are 18 ten‐yen coins in Ichiro’s wallet and just 22 five‐yen coins in his younger brother’s wallet. They decided to place one coin from 

h f h h ff b h deach of them in the offertory box each morning and continue the prayer until either wallet becomes empty. One day they looked into their wallets and found the y ybrother’s amount was bigger than Ichiro’s. How many days since they started prayer?

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TIMSS Video Studies as a Window into Teaching th h P bl S l i (J3)through Problem Solving (J3)

• Pre‐Viewing GuidePre Viewing Guide– What? How? Why?

– Launch, Explore, and SummarizeLaunch, Explore, and Summarize 

– Mathematics: What are the mathematical residues?

– Teaching: What was the role of the teacher? What teaching practices were deployed?

– Learning: What were the roles of the students? How were they engaged?

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Focus on MathematicsFocus on Mathematics

• Important MathematicsImportant Mathematics

• Mathematical Language and Symbolization

h i l S li ( )• Mathematical Storyline (…)

• Mathematical Representations, their sequencing, and their connections

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Focus on TeachingFocus on Teaching

• Pedagogical FlowPedagogical Flow 

• Pedagogical Flexibility

l d f l• Tools and use of Tools

• Questions/Questioning

• Response/Responding

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Focus on LearningFocus on Learning

• Mathematical ResidueMathematical Residue

• Problematization

• Engagement

• Problem Solving Strategies (…)

• Questions/Questioning

• Mathematical LanguageMathematical Language

• Tools and use of Tools

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CommentsComments

•

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Let’s view the videoLet s view the video

• TIMSS Video Studies J3TIMSS Video Studies J3

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Post‐Viewing Think‐PhasePost Viewing Think Phase

• Take 5 minutes to collect your thoughts aboutTake 5 minutes to collect your thoughts about what you have just seen

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Share PhaseShare Phase

• What? How? Why?What? How? Why?– Launch, Explore, and Summarize 

• Mathematics What are the mathematical• Mathematics: What are the mathematical residues?

• Teaching: What was the role of the teacher? What teaching practices were deployed?

• Learning: What were the roles of the students? How were they engaged?

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Focus on MathematicsFocus on Mathematics

• Important MathematicsImportant Mathematics

• Mathematical Language and Symbolization

h i l S li ( )• Mathematical Storyline (…)

• Mathematical Representations, their sequencing, and their connections

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Focus on TeachingFocus on Teaching

• Pedagogical FlowPedagogical Flow 

• Pedagogical Flexibility

l d f l• Tools and use of Tools

• Questions/Questioning

• Response/Responding

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Focus on LearningFocus on Learning

• Mathematical ResidueMathematical Residue

• Problematization

• Engagement

• Problem Solving Strategies (…)

• Questions/Questioning

• Mathematical LanguageMathematical Language

• Tools and use of Tools

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Discussing something,Discussing something, seeing it, and then 

discussing it again is even betterbetter.

– Confucius re‐examined

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What is Teaching Through Problem Solving?What is Teaching Through Problem Solving?

•

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What is Teaching through Problem Solving?What is Teaching through Problem Solving?

• Pose a Problem (What is a problem? versus Exercise?)( p )• Facilitate students’ understanding of the problem• Allow time for students to explore the messiness of the problem and generate conjectures and understandings

• Provide students time to present, share, and examine solution strategies and solutionssolution strategies and solutions

• Summarize the Mathematical Residue• Encourage students to Reflect on their LearningEncourage students to Reflect on their Learning

– Part of this can be tasks/assignments to help students reinforce/elaborate/extend their understanding

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Teaching Through PSTeaching Through PS

• Real World or NOT!• Focus on Methods• Focus on Connections• Focus on Communication• Allow Time• Build Skill as well as Conceptual Understanding

• Facilitate through Questioning• Focus on the Mathematics

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How to Support UnderstandingHow to Support Understanding

• Engage students in solvingEngage students in solving challenging problems

• Examine increasingly better solution• Examine increasingly better solution strategiesP id i f ti f t d t t• Provide information for students at just the right times

i b d– Hiebert and Wearne, 2003, p. 5

• Allow time for students to reflect

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Comments/Questions?Comments/Questions?

•

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Signposts of Teaching Through PSSignposts of Teaching Through PS

• Focus on structure

• Problematization of mathematics for students– posing problems that are within students’ reach, allowing them to struggle 

to find solutions and then examining the methods they have used (Hiebertto find solutions and then examining the methods they have used (Hiebert & Wearne, 2003, p. 6)

– Tasks must allow students to treat the situation as probelmatic, as something they need to think about rather than a prescription they need g y p p yto follow. Secondly, what is problematic about the task should be the mathematics rather than the aspects of the situation. Finally, in order for students to work serious on a task, it must offer students the chance to use kill d k l d h l d (Hi b l 1997 18)skills and knowledge they already possess (Hiebert et al., 1997, p. 18)

• Focus on methods– Recall the offering problem from TIMSS Video Studies J3 Classroom Video

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Focusing on MethodsFocusing on Methods

• Develops mathematical connections amongDevelops mathematical connections among strategies and develops connections among relationships and representations

– Progressively formalize the strategies from concrete to the abstract. Facilitates development of efficient strategiesstrategies.

• Enables error analysis.– Promotes the use of mistakes as sites for– Promotes the use of mistakes as sites for

• Learning

• Addressing misconceptions and preconceptions

• Establishing truth via reason

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Focusing on MethodsFocusing on Methods

• Promotes Equityq y– Progressive formalization of strategies– Multiplicity of views/voices

• Makes students’ ideas visible for public consumption andMakes students  ideas visible for public consumption and reflection.

– Focus on reason and not personages/personality

• Maximizes gains from mathematical struggle– Promotes active learning and student engagement– Promotes metacognition– Provides opportunities to engage in meaningful and worthwhile mathematics

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The Problem of Teaching(Teaching as Problem Solving)

• Can/should tellCan/should tell– Conventions [order of operation, etc.]– Symbolism and representations [tables, graphs, y p [ , g p ,etc.]

– Present and re‐present at times of need

• Can/should present alternative methods to resolve– Those that have not been suggested by students (Hiebert & Wearne, 2003)– Tease out additional strategies for PS toolkit– Can emphasize/label the efficient method as a capstone to exploration

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The Problem of Teaching(Teaching as Problem Solving)

• Can/should/– Highlight information/structures in different methods/strategiesHi hli ht ti th d / t t i– Highlight connections among methods/strategies

– Help students focus on what is important mathematically (i.e., what is the mathematical residue? ‐‐ see Hiebert et al., 1997)

• SummaryMathematical understanding develops over time via– Mathematical understanding develops over time via articulated and coherent “mathematical struggle” negotiated in small steps

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Proficiency in Teaching Mathematics 1Proficiency in Teaching Mathematics 1

• Kilpatrick Swafford & Findell (2001 p 10):Kilpatrick, Swafford, & Findell (2001, p. 10):– Proficiency in teaching mathematics is related to effectiveness: consistently helping students learneffectiveness: consistently helping students learn worthwhile mathematical content. It also entails versatility: being able to work effectively with a wide variety of students in different environments and across a range of mathematical content.

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Kilpatrick, Swafford & Findell (2001) cont.Kilpatrick, Swafford & Findell (2001) cont. 

• page (10):page (10):– Despite the common myth that teaching is little more than common sense or that some peoplemore than common sense or that some people are just born teachers, effective teaching practice can be learned. 

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Kilpatrick, Swafford & Findell (2001) cont (2).Kilpatrick, Swafford & Findell (2001) cont (2).

• page (10):page (10):– Just as mathematical proficiency itself involves interwoven strands, teaching for mathematical proficiency requires similarly interrelated components: conceptual understanding of the core knowledge of mathematics students andcore knowledge of mathematics, students, and instructional practices needed for teaching; procedural fluency in carrying out basic instructional routines; strategic competence in planning  effective instruction and solving problems that arise while teaching;problems that arise while teaching; … 

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Kilpatrick, Swafford, & Findell (2001) cont (3).Kilpatrick, Swafford, & Findell (2001) cont (3).

• page (10):page (10):– adaptive reasoning in justifying and explaining one’s practices and in reflecting on thoseone s practices and in reflecting on those practices; and a productive disposition towards mathematics, teaching, learning, and the improvement of practice.

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