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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)
What is Understanding?What is Understanding?
Understanding:Understanding:
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)
What is Understanding?What is Understanding?
Understanding:Understanding:
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)
What is Understanding?What is Understanding?
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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James & ThomasJames & Thomas
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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