procedural vs conceptual

7/30/2019 Procedural vs Conceptual

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Procedural vs. ConceptualKnowledge: Implications for

Instruction

Presented by:

Dr. David AllenMelisa Jean Hancock


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Consider This Problem Consider the task of learning the multiplication

combination:

7 X 8 = 56

Talk with your partner to come up with asmany good ways as you can to think about theanswer. A good way is one that is simpleenough to become completely mental (nocounting or finger tricks) and should bemathematical.


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The Process of Learning Information Processes Learning Theory

Linked to Behaviorist Theory

The Information Processing Model


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The Process of Learning

Environm

ent

ReceptorsSensoryRegister

Short-termMemory

EffectorsResponseGenerator

Long-TermMemory

Executive Control Expectations

Model employed by information-processing theories of learning and memory. From R.M. Gagne, Essentials of learning for

instruction. Copyright 1974by the Dryden Press, a division of Holt. Rhinehart and Winston. As cited in: Gagne R.M. (1985) TheConditions of Learning 4th Edition. Chicago, Ill.: Holt, Rhinehart, and Winston, Inc.


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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


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No matter how lucidly andpatiently teachers explain totheir students, they cannot

understand for their students.

~Schifter & Fosnot (1993)


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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 aroundthem.

When a teacher demands that students use setmathematical methods, the sense-making activityof students is seriously curtailed.


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Constructivism: Two Major Goals Students should develop mathematical

structures that are more complex, abstract,

and powerful than the ones they currentlypossess so that they are increasingly capableof solving a wide variety of meaningfulproblems.

Students should become autonomous andself-motivated in their mathematical activity.


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Instrumental Understanding


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Relational Understanding


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Consider This Problem Memorize the following string of

numbers:

2 5 8 1 1 1 4 1 7 2 0 2 3


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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


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Elicit ThinkingIn NCTMs Principles of Standards for

School Mathematics, teachers are

encouraged to develop instructionalprograms that enable ALL students to

make and investigate mathematical

conjectures and communicate theirmathematical thinking coherently andclearly.


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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


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Mathematical Knowledge1.Conceptual Knowledge (logical

relationships, representations, an

understanding and ability to talk, write andgive examples of these relationships, etc.)

2. Procedural Knowledge (knowledge ofrules and procedures used in carrying outroutine mathematical tasks and the symbolsused to represent mathematics)


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Conceptual vs. Procedural

Identify who has what kind ofknowledge
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NCTM Standards

The Learning Principle makes it very clearthat learning with UNDERSTANDING is both

essential and possible. That is, ALL childrencan and must learn mathematics withunderstanding. It is impossible to predictthe kinds of problems students will face in

the future. The Learning Principle says thatunderstanding is the only way to ensurethat students will be able to cope with theseunknown problems in the future.


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Unpacking Division

169 14 =


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Procedural Knowledge

It is generally accepted thatprocedural rules should neverbe learned in the absence of a

concept.

(John A. Van De Walle)


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Concrete, Pictorial, Abstract?

Base 10 Blocks (Number Relationships)

Tangrams (Geometric Relationships)

Unifix Cubes (Measures of CentralTendency)

Others, etc. (Share ideas)


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Procedural vs. ConceptualKnowledge: Implications for

Instruction

Presented by:

Dr. David Allen

Melisa Jean Hancock

Copy of presentation available at:

www.educ.ksu.edu/allen
http://www.educ.ksu.edu/allenhttp://www.educ.ksu.edu/allen