Beyond Behaviorism

8/6/2019 Beyond Behaviorism

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Beyond BehaviorismBeyond BehaviorismLearning for UnderstandingLearning for Understanding


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Learning with UnderstandingLearning with Understanding

Entry TaskEntry Task

Individually, take two minutes to writeIndividually, take two minutes to writeyour own definition ofyour own definition ofLearning forLearning forUnderstandingUnderstanding..


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Lesson ObjectivesLesson Objectives

Each student willEach student will Learn that eating candy in class is unacceptableLearn that eating candy in class is unacceptable

Develop and refine a definition ofDevelop and refine a definition ofLearning forLearning forUnderstandingUnderstanding

Compare & contrast student definitions ofCompare & contrast student definitions ofLearning for UnderstandingLearning for Understandingagainst those of leadingagainst those of leadingtheoriststheorists

UtilizeUtilize ZoltansZoltans Stages of LearningStages of Learning Apply theories ofApply theories ofLearning for UnderstandingLearning for Understandingtoto

mathematics instructionmathematics instruction


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William William Dry ReadingDry Reading Brownell (1947) Brownell (1947)

What is Understanding?What is Understanding?

Understanding:Understanding:

Meaning is to be sought in the structure, theMeaning is to be sought in the structure, the

organization, the inner relationships of the subject itself.organization, the inner relationships of the subject itself.

Premise:Premise:

The basic tenant in the proposed instructionalThe basic tenant in the proposed instructionalreorganization is to make arithmetic less a challenge toreorganization is to make arithmetic less a challenge tothe peoples memory and more a challenge to histhe peoples memory and more a challenge to hisintelligence.intelligence.


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Brownell (1945)Brownell (1945)

A LongstandingDebateA LongstandingDebate

It is a mistake to suppose that meaningfulIt is a mistake to suppose that meaningful

arithmetic is something new, somethingarithmetic is something new, somethingcut out of the whole cloth, as it were,cut out of the whole cloth, as it were,during the past twenty or twentyduring the past twenty or twenty--fivefiveyears.years.


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BrownellBrownell

Meanings of vs. Meanings forMeanings of vs. Meanings for Meanings of (pure math)Meanings of (pure math)::

Mathematical understandingsMathematical understandings

Meanings for (applied math)Meanings for (applied math)::

Applications of mathematics to real lifeApplications of mathematics to real life


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BrownellBrownell

Degrees ofMeaningsDegrees ofMeanings Meanings are relative, not absoluteMeanings are relative, not absolute

Degrees of exactness, depth, and growth dependDegrees of exactness, depth, and growth dependon the audienceon the audience

Levels (degrees of meaning) buildLevels (degrees of meaning) build


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BrownellBrownell

Meanings ofArithmeticMeanings ofArithmetic Basic conceptsBasic concepts

(whole numbers, fractions, percents, etc.)(whole numbers, fractions, percents, etc.)

Fundamental operationsFundamental operations(+,(+, -- ,, ,, ))

Principles, relationships, & generalizationsPrinciples, relationships, & generalizations

(e.g. additive identity, commutative property)(e.g. additive identity, commutative property) Decimal number systemDecimal number system


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BrownellBrownell

I

nteractiveI

nterludeI

nteractiveI

nterludeConsider Brownells definition ofLearning forConsider Brownells definition ofLearning for

Understanding:Understanding: Meaning is to be sought in theMeaning is to be sought in the

structure, the organization, the inner relationshipsstructure, the organization, the inner relationshipsof the subject itself.of the subject itself.

Compare and contrast your definition ofCompare and contrast your definition of

Learning for UnderstandingLearning for Understanding(from the entry(from the entrytask) with Brownells.task) with Brownells.


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Zoltan Zoltan MaverickMaverick Dienes (2004) Dienes (2004)Father of the Base 10 BlockFather of the Base 10 Block

What is UnderstandingWhat is Understanding Understanding:Understanding:

ArtArt(an excerpt)(an excerpt)

This is where the artistic creation begins,This is where the artistic creation begins,

Creatures might have no legs or the fish have no fins,Creatures might have no legs or the fish have no fins,

What the brush puts on canvas has been through the fire,What the brush puts on canvas has been through the fire,

Let us get to the meaning we shall all inquire.Let us get to the meaning we shall all inquire.

From the crucible of such creations wild actFrom the crucible of such creations wild act

Comes a language that does not describe any fact !Comes a language that does not describe any fact !

Mental states are thus formed recipients mindsMental states are thus formed recipients minds

So that each one a meaning quite readily finds.

So that each one a meaning quite readily finds.


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ZoltanZoltan

Teacher asSource of KnowledgeTeacher asSource of KnowledgeTeacher delivery of content is not an issueTeacher delivery of content is not an issue

Technology incorporation may be necessary, butTechnology incorporation may be necessary, but

it is not sufficient (i.e.not a silver bullet)it is not sufficient (i

.e.not a silver bullet)

Student may be unable to receive knowledgeStudent may be unable to receive knowledge

Student may be unwilling to receiveStudent may be unwilling to receive

knowledgeknowledgein which case artificial motivationin which case artificial motivationmay be employed (Behaviorism?)may be employed (Behaviorism?)


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ZoltanZoltan

Stages of LearningStages of Learning Initial InteractionInitial Interaction

Discovery of regularities in situations and consequentDiscovery of regularities in situations and consequentplay with sets of rules or restraintsplay with sets of rules or restraints

Comparison of several games possessing the sameComparison of several games possessing the samestructure (search for isomorphisms)structure (search for isomorphisms)

Representation of isomorphic situations in one, allRepresentation of isomorphic situations in one, all--embracing, usually graphical form.embracing, usually graphical form.

Study of the representation by the description of itsStudy of the representation by the description of itsproperties (symbolizing)properties (symbolizing)

Formalization (proof)Formalization (proof)


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ZoltanZoltan

V

ideo MaverickV

ideo Maverick


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ZoltanZoltan

G

ames asStages of LearningG

ames asStages of Learning

Developed games in response to stages ofDeveloped games in response to stages of

learninglearning

Preliminary gamesPreliminary games

Structured gamesStructured games

Practice gamesPractice games


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ZoltanZoltan

I

nteractiveI

nterludeI

nteractiveI

nterludeCircular VillagesCircular Villages

Organize yourselves into groups of 3Organize yourselves into groups of 3 4. Your4. Yourteacher will distribute theteacher will distribute the CircularVillagesCircularVillages

handout. Use Zoltanshandout. Use Zoltans Stages of LearningStages of Learningtoto

complete steps 1complete steps 1 3. Raise your hand if you3. Raise your hand if youneed assistance and follow all directionsneed assistance and follow all directions

carefully or there will be no break.carefully or there will be no break.


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Jerome Jerome RenegadeRenegade Bruner (1996) Bruner (1996)



Is the outcome of organizing and contextualizing essentiallyIs the outcome of organizing and contextualizing essentiallycontestable, incompletely verifiable propositions in a disciplinedcontestable, incompletely verifiable propositions in a disciplinedway.way.


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BrunerBruner

B

runersC

onceptual FrameworkB

runersC

onceptual FrameworkA person categorizes new objects andA person categorizes new objects and

events that occur in his environmentevents that occur in his environment

according to the properties they areaccording to the properties they areseen to have in common with otherseen to have in common with otherobjects and events previouslyobjects and events previouslycategorized.categorized.

Warp to Forms of RepresentationWarp to Forms of Representation


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BrunerBruner

Bruner in HistoricalContextBruner in HistoricalContext

1957: USSR launches Sputnik1957: USSR launches Sputnik

1959:1959: Woods HoleConferenceWoods HoleConferenceononcurriculum reformcurriculum reform

1960:1960: The Process ofEducationThe Process ofEducation


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Bruner (1960)Bruner (1960)

The Process ofEducationThe Process ofEducation

Any subject can be taughtAny subject can be taught

effectively in some intellectuallyeffectively in some intellectuallyhonest form to any child at anyhonest form to any child at any

stage of development.stage of development.


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BrunerBruner

Forms of RepresentationForms of Representation Enactive modeEnactive mode

Involves human motor capacitiesInvolves human motor capacities Iconic modeIconic mode

depends on visual or sensory organizationdepends on visual or sensory organization Symbolic modeSymbolic mode

involves reasoning, words, and languageinvolves reasoning, words, and language

New forms are added and become dominant, but oldNew forms are added and become dominant, but oldforms remainforms remain

Skip to Bruner InterludeSkip to Bruner Interlude


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BrunerBruner

SpiralCurriculumSpiralCurriculum

Teach a meaningful topic at a levelTeach a meaningful topic at a levelappropriate for the child and revisitappropriate for the child and revisit

later to create a more explicit andlater to create a more explicit andmature understanding.mature understanding.


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BrunerBruner

InteractiveInterludeInteractiveInterlude

In groups of 3In groups of 3 4, compare and contrast each4, compare and contrast each

others definitions ofothers definitions ofLearning with UnderstandingLearning with Understanding.

.

Discuss each others definitions in light of thoseDiscuss each others definitions in light of thoseofBruner, Zoltan, and Brownell.ofBruner, Zoltan, and Brownell.


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Richard Richard BuzzwordBuzzword Skemp Skemp

(1971)(1971)



To understand something means to assimilate it into anTo understand something means to assimilate it into anappropriate schema.appropriate schema.(schema: a conceptual structure)(schema: a conceptual structure)

Premise:Premise:

The widespread negative attitude towards mathematicsThe widespread negative attitude towards mathematics

is caused by the widespread failure to teach relationalis caused by the widespread failure to teach relationalmathematics.mathematics.


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Skemp (1977)Skemp (1977)

Instrumental & Relational UnderstandingInstrumental & Relational Understanding

Instrumental UnderstandingInstrumental Understanding::

Possessing a rule and the ability use itPossessing a rule and the ability use itrulesruleswithout reasonswithout reasons

Relational UnderstandingRelational Understanding::Knowing both what to do and whyKnowing both what to do and why


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SkempSkemp

Example ofInstrumental & RelationalExample ofInstrumental & Relational

UnderstandingUnderstanding Instrumental UnderstandingInstrumental Understanding: Justification by FOIL: Justification by FOIL

(a+ b)2 = (a+ b)(a+ b) = a2 +2ab+ b2

Relational Understanding: Picture proof

a

b

a b

a2

b2ab

ab


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SkempSkemp

InteractiveInterlude PartIInteractiveInterlude PartI

In groups of 2, use Skemps definition ofIn groups of 2, use Skemps definition of

Instrumental Understanding (Instrumental Understanding (possessing a rule andpossessing a rule andthe ability use itthe ability use itrules without reasons)rules without reasons) to identify,to identify,

list, and discuss mathematics examples oflist, and discuss mathematics examples oftextbook and classroom instrumentaltextbook and classroom instrumental

explanations.explanations.


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SkempSkemp

InteractiveInterlude PartIIInteractiveInterlude PartII

In groups of 2, use Skemps definition ofIn groups of 2, use Skemps definition ofRelational Understanding (Relational Understanding (knowing both whatknowing both what

to do and why)to do and why) toto attempt to provideattempt to providerelational explanations for the traditionalrelational explanations for the traditionalalgorithms for dividing two fractions or foralgorithms for dividing two fractions or for

finding the area of a circle (your choice).

finding the area of a circle (your choice).

Warp toWarp to HiebertHiebert & Carpenter& Carpenter


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SkempSkemp

Instrumental/RelationalMismatchesInstrumental/RelationalMismatches

Instrumental Students + Relational Teacher =Instrumental Students + Relational Teacher =

Frustrated TeacherFrustrated Teacher

Relational Students + Instrumental Teacher =Relational Students + Instrumental Teacher =Frustrated Student (negative, farFrustrated Student (negative, far--reachingreachingconsequences)consequences)

Instrumental Teacher + Relational Textbook =Instrumental Teacher + Relational Textbook =Frustrated AuthorFrustrated Author


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James James AbbotAbbot Hiebert & Hiebert &

Thomas Thomas CostelloCostello Carpenter Carpenter

(2000)(2000)


Understanding:

Understanding:

We understand something if we see how it is related or connected toWe understand something if we see how it is related or connected toother things we know.other things we know.

(Hiebert cites Brownell, Carpenter & Hiebert)(Hiebert cites Brownell, Carpenter & Hiebert)

Premise:Premise:

Understanding is theUnderstanding is the most fundamental goalmost fundamental goalof mathematicsof mathematicsinstruction (emphases added)instruction (emphases added)


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James & ThomasJames & Thomas

Importance of UnderstandingImportance of Understanding

ToTo really knowreally knowmath requires that it ismath requires that it is

understood; utility is not enoughunderstood; utility is not enough

C

onfidenceC

onfidence EngagementEngagement

Internally rewarding/satisfying (compare toInternally rewarding/satisfying (compare to

B

ehaviorism)B

ehaviorism)


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The Path to UnderstandingThe Path to Understanding

ReflectingReflecting(origins in cognitive psychology,(origins in cognitive psychology,

emphasis on internal mental operations)emphasis on internal mental operations)

CommunicatingCommunicating(origins in social cognition with(origins in social cognition with

emphasis on the context of learning and socialemphasis on the context of learning and socialinteraction)interaction)

Warp to Pros & ConsWarp to Pros & Cons OROR Warp to Exit TaskWarp to Exit Task


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Classroom FrameworkClassroom Framework

Reflecting & communicating are crucial. WeReflecting & communicating are crucial. We

need a framework within which these canneed a framework within which these canhappen.happen.

Classroom tasksClassroom tasks

Teachers RoleTeachers Role

C

lassroom social cultureC

lassroom social culture Mathematical toolsMathematical tools

Equity and accessibilityEquity and accessibility


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Framework 1Framework 1Classroom TasksClassroom Tasks

Make math problematicMake math problematic

Connect with students where they are at (zoneConnect with students where they are at (zone

of proximal development)of proximal development)

Leave behind something of mathematical valueLeave behind something of mathematical value(usefulness is important [applied math] but pure(usefulness is important [applied math] but pure

mathematics must also result)mathematics must also result)


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Framework 2Framework 2Teachers RoleTeachers Role

Ideas & methods must be valuedIdeas & methods must be valued

Share essential information (not too hot, not tooShare essential information (not too hot, not too

cold, jusssst right)cold, jusssst right)

Establish classroom cultureEstablish classroom culture


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Framework 3Framework 3ClassroomCultureClassroomCulture

Ideas & methods must be valuedIdeas & methods must be valued

Students choose & share methodsStudents choose & share methods

Mistakes are learning opportunitiesMistakes are learning opportunities

Correctness resides in mathematical argumentCorrectness resides in mathematical argument


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Framework 4Framework 4Mathematics ToolsMathematics Tools

Meaning of tools must be constructed by eachMeaning of tools must be constructed by each

studentstudent

Used to solve problemsUsed to solve problems

Used to record, communicate, & thinkUsed to record, communicate, & think


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Framework 5Framework 5Equity &AccessibilityEquity &Accessibility

Tasks accessible to all studentsTasks accessible to all students

Each student has a voice in the classroomEach student has a voice in the classroom

Every student contributesEvery student contributes

Warp to exit taskWarp to exit task


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Rote vs. ConceptualRote vs. Conceptual

Pros &ConsPros &ConsRoteRote

Within its own context,Within its own context,instrumental mathematics isinstrumental mathematics isusually easier to understand.usually easier to understand.

Rewards are more immediateRewards are more immediate& more apparent& more apparent

One can get the correctOne can get the correctanswer quickly and reliablyanswer quickly and reliably

Harder to retain (in absenceHarder to retain (in absence

of repetition)of repetition)

ConceptualConceptual

TransferableTransferable

Easier to remember/recoverEasier to remember/recover

Harder to learnHarder to learn

Relational knowledge can beRelational knowledge can beeffective as a goal in itselfeffective as a goal in itself

Relational schemas areRelational schemas areorganic in qualityorganic in quality

Reduces repetitive practiceReduces repetitive practice

Safeguards against sillySafeguards against sillyanswersanswers

Versatility of attackVersatility of attack


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Teaching for UnderstandingTeaching for Understanding

GeneralGeneralImplementationIssuesImplementationIssues

Standardized/Required TestingStandardized/Required Testing

Breadth of coverageBreadth of coverage

Assessment difficultiesAssessment difficulties

Teacher acceptanceTeacher acceptance


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Teaching for UnderstandingTeaching for Understanding

RuralImplementationIssuesRuralImplementationIssues

Good ole boysGood ole boys

Good enough for meGood enough for me good enough for mygood enough for my

kidskids Change is badChange is bad

Lack of resources (e.g. technology)Lack of resources (e.g. technology)

Homework supportHomework support Projects that require outProjects that require out--ofof--class researchclass research


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Final InterludeFinal Interlude

Exit TaskExit Task

Individually,Individually, refinerefineandand explainexplainyour originalyour originaldefinition ofdefinition ofLearning with UnderstandingLearning with Understanding. If you. If youbelieve that your original definition does notbelieve that your original definition does not

need to be refined, please explain why.need to be refined, please explain why.

When you are done, you may be excused ifWhen you are done, you may be excused if

you can demonstrate, via possession of atyou can demonstrate, via possession of atleast 5 packages ofleast 5 packages ofSmarties,Smarties, that youthat youparticipated in todays class.participated in todays class.


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ReferencesReferences

Bart, W. (1970). Mathematics Education: The Views of Zoltan Dienes.Bart, W. (1970). Mathematics Education: The Views of Zoltan Dienes. TheSchool Review, Vol. 78, No. 3TheSchool Review, Vol. 78, No. 3..

Brownell, W. (1945). The Natural Sciences and Mathematics.Brownell, W. (1945). The Natural Sciences and Mathematics. Review ofEducation Research, Vol. 15, No. 4Review ofEducation Research, Vol. 15, No. 4..

Brownell, W. (1947). The Place of Meaning in the Teaching ofArithmetic.Brownell, W. (1947). The Place of Meaning in the Teaching ofArithmetic.TheElementarySchool JournalTheElementarySchool Journal, Vol., Vol.47, No. 5.47, No. 5.

Bruner, J. (1966).Bruner, J. (1966).Toward a Theory ofInstructionToward a Theory ofInstruction. W.W. Norton & Company, New York.. W.W. Norton & Company, New York.

Bruner, J. (1966).Bruner, J. (1966).TheCulture ofEducationTheCulture ofEducation. Harvard UniversityPress, Cambridge, MA.. Harvard UniversityPress, Cambridge, MA.

Bruner, J. (1966).Bruner, J. (1966).The Process ofEducationThe Process ofEducation. Harvard UniversityPress, Cambridge, MA.. Harvard UniversityPress, Cambridge, MA.

Dienes, Z. (1960).Dienes, Z. (1960).Building Up MathematicsBuilding Up Mathematics. Hutchinson Educational LTD, London.. Hutchinson Educational LTD, London. Hiebert, J, Carpenter, T, & others. (1996).Problem Solving as a Basis for Reform in Curriculum andHiebert, J, Carpenter, T, & others. (1996).Problem Solving as a Basis for Reform in Curriculum and

Instruction: The Case of Mathematics.Instruction: The Case of Mathematics. Educational Researcher, Vol. 25, No. 4.Educational Researcher, Vol. 25, No. 4.

Hiebert, J, Carpenter, T, & others. (2000).Hiebert, J, Carpenter, T, & others. (2000). MakingSense: teaching and learning mathematics with understandingMakingSense: teaching and learning mathematics with understanding..

Kilpatrick, J., Wearver, J.F. (1977). The Place of William A.Brownell in Mathematics Education.Kilpatrick, J., Wearver, J.F. (1977). The Place of William A.Brownell in Mathematics Education.Journal forJournal forResearch inMathematicsEducation, Vol. 8, No. 5Research inMathematicsEducation, Vol. 8, No. 5

Noddings, N. (1994). William Brownell and The Search for Meaning.Noddings, N. (1994). William Brownell and The Search for Meaning.Journal for Research inMathematicsJournal for Research inMathematicsEducation, Vol. 24, No. 5Education, Vol. 24, No. 5..

Schoenfeld, A. (1992).Learning to Think Mathematically: Problem Solving, Metacognition, and SenseSchoenfeld, A. (1992).Learning to Think Mathematically: Problem Solving, Metacognition, and Sense--MakingMakingin Mathematics.in Mathematics. Handbook on Research forMathematics Teaching and LearningHandbook on Research forMathematics Teaching and Learning..

Schoenfeld, A. (2005).Schoenfeld, A. (2005).Mathematics Teaching and LearningMathematics Teaching and Learning. University ofCalifornia, Berkeley.. University ofCalifornia, Berkeley.

Skemp, R. (1971).Skemp, R. (1971).Psychology of LearningMathPsychology of LearningMath.Penguin Books Ltd, Harmondsworth..Penguin Books Ltd, Harmondsworth.

Skemp, R. (1976).Skemp, R. (1976).Relational Understanding andInstrumental UnderstandingRelational Understanding andInstrumental Understanding.. Mathematics Teaching, 77Mathematics Teaching, 77..

Skemp, R. (1987).Skemp, R. (1987).The Psychology of KnowingMathThe Psychology of KnowingMath.Lawrence Erlbaum Associates. Hillsdale, NJ..Lawrence Erlbaum Associates. Hillsdale, NJ.

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