UAS

UAS

MathematicalProficiency

ConceptualUnderstanding

DOINGMATH

ProblemSolving

ProceduralSkills

Problem Solving

Computational & Procedural Skills

DOING MATH

Conceptual Understanding

“WHERE” THEMATHEMATICSWORKS

“HOW” THE

MATHEMATICSWORKS

“WHY” THE

MATHEMATICSWORKS

UAS

The“WHAT”

• ConceptualUnderstanding

• interpretstheconceptsrelatedtothetaskandtranslatesthemintomathematicalideas

• usesmathematicsthatfitwhatisrequestedintheproblem

UAS

The“HOW”

• Strategies/approachesforsolvingproblems• choosesandcarriesoutprocessesandstrategiesthatcanwork.

• usesappropriatepictures,models,diagrams,and/orsymbols.

UAS

The“WHY”

• VerificationReasoningandCommunication• “defense”insolvingthetask• clearlyshowsthereviewofconcepts,processes,andcalculationsusedtogetthesolution.

• clearlyshowsthepathleadingtothesolutionshownwithnogapsforthereadertofillin.

• workfitsallthepartstogetherbyusingpictures,charts,diagrams,and/orwords.

UAS

Arithmetic

• “thestudyofnumbers,especiallythepropertiesofthetraditionaloperationsbetweenthem— addition,subtraction,multiplicationanddivision.”

UAS

Mathematics

Mathematicsisawayofthinkingabout,understanding,explaining,andexpressing

phenomena.

UAS

Mathematicsisaboutinquiryandinsight.Computationis(usually)ameanstoanend.

Problem solving is at the core of mathematician’s work!

A paradigm shift in Instructional Practice

UAS

6 Shifts in Mathematics

FocusCoherenceFluencyDeep UnderstandingApplicationsDual Intensity

UAS

Lessons Learned• Mile wide and inch deep does not work.

• The task ahead is not so much about how many specific topics are taught; rather, it is more about ways of thinking.

• To change students’ ways of thinking, we must change how we teach.

UAS

Lesson ComparisonUnited States and Japan

The emphasis on skill acquisition is evident in the steps most common in U.S. classrooms

The emphasis on understanding is evident in the steps of a typical Japanese lesson

•Teacher instructs students in concept or skill

•Teacher solves problems with/for the class

•Students practice on their own while teacher assists individual students

•Teacher poses a thought provoking problem

•Students and teachers explore the problem

•Various students present ideas or solutions to the class

•Teacher summarizes the class solutions

•Students solve similar problems

12UAS

Effective Mathematics Teaching Practices

1. Establish mathematics goals to focus learning.2. Implement tasks that promote reasoning and problem

solving. 3. Use and connect mathematical representations.4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions.6. Build procedural fluency from conceptual understanding.7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking.

UAS

UAS

StandardsforMathematicalPractice

1.Makesenseofproblemsandpersevereinsolvingthem.2.Reasonabstractlyandquantitatively.3.Constructviableargumentsandcritiquethereasoningofothers.

4.Modelwithmathematics.5.Useappropriatetoolsstrategically.6.Attendtoprecision.7.Lookforandmakeuseofstructure.8.Lookforandexpressregularityinrepeatedreasoning.

Describe ways students engage with the subject matter throughout the elementary, middle, and high school years

ProblemsandProblemSolving

“Most,ifnotall,importantmathematicsconceptsandprocedurescanbestbetaughtthroughproblemsolving.”

--JohnVandeWalle

UAS

Whatdoesitlooklikeinthetasksweaskchildrentodo?We do not “teach” the practices,rather we design our instruction around tasks that give students the opportunity to use and develop the practices in learning and doing mathematics everyday, not only in mathematics class but in all aspects of their education and lives that include mathematical thinking.

UAS

UAS

UAS

UAS

Implement Tasks that Promote Reasoning and Problem Solving

• Mathematical tasks should:

• Provide opportunities for students to engage in exploration or encourage students to use procedures in ways that are connected to concepts and understanding;

• Build on students’ current understanding; and

• Have multiple entry points.

• There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

• (Lappan & Briars, 1995)

UAS

• Teachingfor problemsolving• teachingskills,thenproviding

problemstopracticethoseskills

• Teachingabout problemsolving• teachingstrategies

• Teachingthrough problemsolving• teachingcontentthroughproblems• posingquestionsandinvestigationsas

keytolearningnewmathematics(beginningaunitofworkwithaproblemthestudentscannotdoyet)

• “Most,ifnotall,importantmathematicsconceptsandprocedurescanbestbetaughtthroughproblemsolving.”

• --JohnVandeWalle

UAS

Polya (from a lecture on teaching)

• Mathematics is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean to be doing mathematics? In the first place, it means to be able to solve mathematical problems.”

UAS

AShift

Awayfrom:• Explain,practice,thenstoryproblems• Oneway(theteacher’sway)• Showandtellapproachwiththestudentasapassivelearner• Problem-solvingproblemsasaseparateactivity• Expectingexplicitdirectionsforhowtosolveproblems• Studentslearnmathematicsasaresultofproblemsolving.• Childrenarelearningmathematicsbydoingit!

UAS

• Whatisaproblem?• “Aproblemisataskthatthe

individualwantstoachieve,andforwhichheorshedoesnothaveaccesstoastraightforwardmeansofsolution.”

(Schoenfeld,1985)

• “....problemsshouldrelatebothtotheapplicationofmathematicstoeverydaysituationswithinthepupils'experience,andalsotosituationswhichareunfamiliar.”

• (Cockcroft,1982,para249)

• WhatisProblemSolving?• “Problemsolvingmeansengaginginataskforwhichthesolutionmethodisnotknowninadvance.”--PrinciplesandStandardsfor

SchoolMathematics

• Itencompassesexploring,reasoning,strategizing,estimating,conjecturing,testing,explaining,andproving.

UAS

ProblemorExercise?

• Anexerciseisasetofnumbersentencesintendedforpracticeinthedevelopmentofaskill.

• Aproblemiswhatwecommonlyrefertoasa“wordproblem.”• Butbeware!• Problemscanbecomeexercises!!

UAS

• Willisearns$11aweekmowinglawns.Hewantstoearnenoughmoneytobuyavideogamesystemthatcosts$110.

• FindtheactualnumberofweeksitwilltakeWillistoearnenoughmoneytobuythevideogamesystem.

• Willisearns$10-25aweekmowinglawns.Hewantstoearnenoughmoneytobuyavideogamesystemthatcosts$150.

• Giveaminimumof3waysinwhichhecanearnthemoneytopurchasethegamesystem.(Showwork)

• Whichwaywouldbethemostefficientwaytoearnthemoney?Explain.

UAS

CommonCharacteristicsofaRICHPROBLEM

• Itshouldbechallenging tothelearner.• Itshouldholdthelearner’s interest.• Thelearnershouldbeabletoconnect theproblemtoherlifeand/ortoothermathproblemsorsubjects.

• Itshouldcontainarangeofchallenges.• Itshouldbeabletobesolvedinseveralways.

UAS

TheValueofTeachingThroughProblemSolving

• Thefocusofthestudents’attentiononideasandsensemaking• Developsthebeliefinstudentsthattheyarecapableofdoingmathematicsandthatmathematicsmakessense!

• Providesacontexttohelpstudentsbuildmeaningfortheconcept• Allowsanentrypointforawiderangeofstudents• Providesongoingassessmentdatathatcanbeusedtomakeinstructionaldecisions,helpstudentssucceed,andinformparents

• Allowsforextensionsandelaborations• Engagesstudentssothattherearefewerclassroomdisciplineproblems• Develops“mathematicalpower”• Itisalotoffun!

UAS

ChoosingProblem-SolvingTasks

• Theproblemmustbemeaningfultothestudents.• Theteachermustsometimesadapttheproblemtomakeitmoremeaningful.

• Theteachermustworktheproblemtoanticipatemathematicalideasandpossiblequestionsthatproblemmightraise.

UAS

Food for thought….① We are being asked to teach in distinctly

different ways from how we were taught.② The traditional curriculum was designed to

meet societal needs that no longer exist.③ It is unreasonable to ask a professional to

change more than 10 percent a year, but it is unprofessional to change by much less than 10 percent a year.

http://steveleinwand.com/wp-content/uploads/2014/08/FourPostulatesforChange.pdf

UAS