Does C + P = P + C ?Does C + P = P + C ?Conceptual versus Procedural Conceptual versus Procedural

UnderstandingUnderstanding

Dr. Jeremy WintersDr. Jeremy Winters

ElEd 6550ElEd 6550

Psychological Considerations to Psychological Considerations to Teaching MathTeaching Math

Mathematics should beMathematics should beA natural outgrowth of the A natural outgrowth of the

children’s liveschildren’s lives Interesting for the studentsInteresting for the studentsChallenge their imaginationChallenge their imaginationBeget creative solutionsBeget creative solutions

Logical ApproachLogical Approach

Usual OrderUsual Order

This approach does not ensure student This approach does not ensure student success.success.

Need to complement with Psychological Need to complement with Psychological approaches.approaches.

Psychological ApproachesPsychological Approaches

Piaget: Stages of DevelopmentPiaget: Stages of Development1. Sensorimotor (0-2 years)1. Sensorimotor (0-2 years)

Imitate sounds and actions, and recognizes that Imitate sounds and actions, and recognizes that objects still exist when they are out of sight.objects still exist when they are out of sight.

2. Preoperational(2-7 years)2. Preoperational(2-7 years)Child gains an initial use of language and the Child gains an initial use of language and the

ability to think in symbolic termsability to think in symbolic terms

Psychological ApproachesPsychological Approaches

Piaget: Stages of DevelopmentPiaget: Stages of Development3. Concrete Operations (7-11 years)3. Concrete Operations (7-11 years)

Physical objects provide the medium for learningPhysical objects provide the medium for learningChildren discovery objects can be changed or Children discovery objects can be changed or

moved and still retain many of their characteristics moved and still retain many of their characteristics and that these changes can be reversedand that these changes can be reversed

4. Formal Operations (around 11)4. Formal Operations (around 11)Adults may never fully operate at this levelAdults may never fully operate at this levelStudents can think logically about abstract Students can think logically about abstract

problems.problems.

Piagetian TermsPiagetian Terms

Conservation (age 7)Conservation (age 7)

Assimilation and AccommodationsAssimilation and Accommodations

Children should be involved in inventing Children should be involved in inventing mathematics. It is through experiences mathematics. It is through experiences that children discover relationships and that children discover relationships and solve problems. (Constructivism)solve problems. (Constructivism)

Hot Air BalloonsHot Air Balloons

Your entryYour entry

Things I want you to knowThings I want you to know

ExamplesExamples

Students at Scales Elementary SchoolStudents at Scales Elementary School

Conceptual versus Procedural Conceptual versus Procedural KnowledgeKnowledge

What are they?What are they?

From the Hot Air Balloon ActivityFrom the Hot Air Balloon Activity

““Students taught in a way that relies too heavily on rote Students taught in a way that relies too heavily on rote memorization isolated from meaning have difficulty memorization isolated from meaning have difficulty

recovering and retaining math concepts and recovering and retaining math concepts and generalizations.” - Shermangeneralizations.” - Sherman

Does C + P = P + C?, notice neither is aloneDoes C + P = P + C?, notice neither is alone

ConceptsConcepts

Concepts can not be known until they are Concepts can not be known until they are experienced.experienced.

Conceptual versus Procedural Conceptual versus Procedural KnowledgeKnowledge

What Barney has to SayWhat Barney has to Say

The Dissimilar LearnerThe Dissimilar Learner

Long-term understanding and skill Long-term understanding and skill achievement are established together achievement are established together

when students successively build upon when students successively build upon concepts in a guided discovery process.concepts in a guided discovery process.

The Dissimilar LearnerThe Dissimilar LearnerMathematics LearningMathematics Learning5 Interrelated Strands5 Interrelated Strands

1.1. Conceptual UnderstandingConceptual UnderstandingComprehension of ideasComprehension of ideas

2.2. Procedural FluencyProcedural FluencyFlexible and accurate skills and proceduresFlexible and accurate skills and procedures

3.3. Strategic CompetenceStrategic CompetenceAbility to formulate and solve problemsAbility to formulate and solve problems

4.4. Adaptive ReasoningAdaptive ReasoningCapacity to reflect and evaluate one’s knowledge and ability Capacity to reflect and evaluate one’s knowledge and ability

to reasonto reason

5.5. Productive DispositionProductive DispositionHabitual inclination to make sense of and value what is being Habitual inclination to make sense of and value what is being

learned learned

The Dissimilar LearnerThe Dissimilar Learner

One who has experienced little or no One who has experienced little or no success in all five areas or lack any success in all five areas or lack any understanding in one complete area.understanding in one complete area.

The Dissimilar LearnerThe Dissimilar LearnerDesigning Lessons for SuccessDesigning Lessons for Success

1.1. Connect new concepts to those the students Connect new concepts to those the students know and are actively engaged in at a know and are actively engaged in at a concrete level of understandingconcrete level of understanding

2.2. Students represent understanding with Students represent understanding with pictures or diagramspictures or diagrams

3.3. Students attach numerals and number Students attach numerals and number sentences to the drawingssentences to the drawings

4.4. Students practice skills and algorithmic Students practice skills and algorithmic procedures through a variety of activities and procedures through a variety of activities and reinforcement lessons.reinforcement lessons.