Does C + P = P + C ? Conceptual versus Procedural Understanding Dr. Jeremy Winters ElEd 6550

Psychological Considerations to Teaching Math Mathematics should be Mathematics should be A natural outgrowth of the children’s lives Interesting for the students Challenge their imagination Beget creative solutions

Logical Approach Usual Order Usual Order This approach does not ensure student success. This approach does not ensure student success. Need to complement with Psychological approaches. Need to complement with Psychological approaches.

Psychological Approaches Piaget: Stages of Development Piaget: Stages of Development 1. Sensorimotor (0-2 years) Imitate sounds and actions, and recognizes that objects still exist when they are out of sight. Imitate sounds and actions, and recognizes that objects still exist when they are out of sight. 2. Preoperational(2-7 years) Child gains an initial use of language and the ability to think in symbolic terms Child gains an initial use of language and the ability to think in symbolic terms

Psychological Approaches Piaget: Stages of Development Piaget: Stages of Development 3. Concrete Operations (7-11 years) Physical objects provide the medium for learning Physical objects provide the medium for learning Children discovery objects can be changed or moved and still retain many of their characteristics and that these changes can be reversed Children discovery objects can be changed or moved and still retain many of their characteristics and that these changes can be reversed 4. Formal Operations (around 11) Adults may never fully operate at this level Adults may never fully operate at this level Students can think logically about abstract problems. Students can think logically about abstract problems.

Piagetian Terms Conservation (age 7) Conservation (age 7) Assimilation and Accommodations Assimilation and Accommodations Children should be involved in inventing mathematics. It is through experiences that children discover relationships and solve problems. (Constructivism) Children should be involved in inventing mathematics. It is through experiences that children discover relationships and solve problems. (Constructivism)

Hot Air Balloons Your entry Things I want you to know Examples Students at Scales Elementary School

Conceptual versus Procedural Knowledge What are they? From the Hot Air Balloon Activity “Students taught in a way that relies too heavily on rote memorization isolated from meaning have difficulty recovering and retaining math concepts and generalizations.” - Sherman Does C + P = P + C?, notice neither is alone

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

Conceptual versus Procedural Knowledge What Barney has to Say

The Dissimilar Learner Long-term understanding and skill achievement are established together when students successively build upon concepts in a guided discovery process.

The Dissimilar Learner Mathematics Learning 5 Interrelated Strands 1. Conceptual Understanding Comprehension of ideas 2. Procedural Fluency Flexible and accurate skills and procedures 3. Strategic Competence Ability to formulate and solve problems 4. Adaptive Reasoning Capacity to reflect and evaluate one’s knowledge and ability to reason 5. Productive Disposition Habitual inclination to make sense of and value what is being learned

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

The Dissimilar Learner Designing Lessons for Success 1. Connect new concepts to those the students know and are actively engaged in at a concrete level of understanding 2. Students represent understanding with pictures or diagrams 3. Students attach numerals and number sentences to the drawings 4. Students practice skills and algorithmic procedures through a variety of activities and reinforcement lessons.