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Procedural vs. Conceptual Knowledge: Implications for Instruction

Consider This Problem. Consider the task of learning the multiplication combination:7 X 8 = 56 Talk with your partner to come up with as many good ways as you can to think about the answer. A good way is one that is simple enough to become completely mental (no counting or finger tricks) and should be mathematical..

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Procedural vs. Conceptual Knowledge: Implications for Instruction

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    1. Procedural vs. Conceptual Knowledge: Implications for Instruction Presented by: Dr. David Allen Melisa Jean Hancock

    2. Consider This Problem Consider the task of learning the multiplication combination: 7 X 8 = 56 Talk with your partner to come up with as many good ways as you can to think about the answer. A good way is one that is simple enough to become completely mental (no counting or finger tricks) and should be mathematical.

    3. The Process of Learning Information Processes Learning Theory Linked to Behaviorist Theory The Information Processing Model

    4. The Process of Learning

    5. Traditional Geometry Lesson Circles: Formulas and Definitions Practice Short Term Memory Practice Short ?Long Term memory Practice Long Term Memory Practice Long Term Memory Practice Permanent Memory Assessssssssssss and Move on

    6. No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students.

    7. Basic Tenets of Constructivism Knowledge is actively created or invented by the child Children create new mathematical knowledge by reflecting on their physical and mental actions. Learning is a social process in which children grow into the intellectual life of those around them. When a teacher demands that students use set mathematical methods, the sense-making activity of students is seriously curtailed.

    8. Constructivism: Two Major Goals Students should develop mathematical structures that are more complex, abstract, and powerful than the ones they currently possess so that they are increasingly capable of solving a wide variety of meaningful problems. Students should become autonomous and self-motivated in their mathematical activity.

    9. Instrumental Understanding

    10. Relational Understanding

    11. Consider This Problem Memorize the following string of numbers: 2 5 8 1 1 1 4 1 7 2 0 2 3

    12. Implications for Teaching and Learning: What does a Constructivist teacher do? Create a mathematical environment Pose worthwhile (cognitively demanding) mathematical tasks Use cooperative learning groups Use models and calculators as thinking tools Require justification of student responses Encourage discourse and writing Listen actively

    13. Elicit Thinking In NCTM�s Principles of Standards for School Mathematics, teachers are encouraged to develop instructional programs that enable ALL students to �make and investigate mathematical conjectures� and �communicate their mathematical thinking coherently and clearly.�

    14. Constructivist Geometry Lesson Discovering geometry relationships existing within the construct of a circle Investigate and explore Collect Data and Analyze Make conjectures and generalizations Assess understanding and move on

    15. Mathematical Knowledge 1.Conceptual Knowledge (logical relationships, representations, an understanding and ability to talk, write and give examples of these relationships, etc.) 2. Procedural Knowledge (knowledge of rules and procedures used in carrying out routine mathematical tasks and the symbols used to represent mathematics)

    16. Conceptual vs. Procedural Identify who has what kind of knowledge

    17. NCTM Standards The Learning Principle makes it very clear that learning with UNDERSTANDING is both essential and possible. That is, ALL children can and must learn mathematics with understanding. It is impossible to predict the kinds of problems students will face in the future. The Learning Principle says that understanding is the only way to ensure that students will be able to cope with these unknown problems in the future.

    18. Unpacking Division 169 � 14 =

    19. Procedural Knowledge It is generally accepted that procedural rules should never be learned in the absence of a concept. (John A. Van De Walle)

    20. Procedural vs. Conceptual Knowledge Objects and names of objects are not the same as relationships between objects.

    21. Concrete, Pictorial, Abstract? Base 10 Blocks (Number Relationships) Tangrams (Geometric Relationships) Unifix Cubes (Measures of Central Tendency) Others, etc. (Share ideas)

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