EEX 4251EEX 4251

Developing Mathematical Developing Mathematical KnowledgeKnowledge

Dr. Matthew C. Robinson, Spring 2008

The Math LessonThe Math Lesson

33 4 4 11 11 12 12 13 13+ 1+ 1 + 2+ 2 + 1+ 1 + 2+ 2 + 4+ 4 44 1111 1212 1414 102102

101101 101 101 113 113 1001 1001 100 100+ 10+ 10 + 4+ 4 + 14+ 14 + 201+ 201 +110+110 111111 110110 10021002 12021202 10001000

The LessonThe Lesson

101101 1004 1004 203 203 1002 1002 1013 1013+202+202 + 103+ 103 +114+114 + 113+ 113 + 101+ 101

104104 112 112 13 13 112 112 104 104+201+201 +202+202 +101+101 + 104+ 104 +203+203

PracticePractice

10131013 1112 1112 1112 1112 1200 1200 1114 1114

11001100 1104 1104 114 114 10111011 11021102

Constructivist Constructivist TheoriesTheories

From:From:

Clements, Douglas H. and Michael Battista. Clements, Douglas H. and Michael Battista. “Constructivist Learning and Teaching” “Constructivist Learning and Teaching” The The Arithmetic TeacherArithmetic Teacher, September 1990, pp. 34-35., September 1990, pp. 34-35.

Knowledge is actively created Knowledge is actively created or invented by the child, not or invented by the child, not passively received from the passively received from the environment.environment.

Children create new Children create new mathematical knowledge by mathematical knowledge by reflecting on their physical and reflecting on their physical and mental actions. Ideas are mental actions. Ideas are constructed or made meaningful constructed or made meaningful when children integrate them when children integrate them into their existing structures of into their existing structures of knowledge.knowledge.

No one true reality exists, only No one true reality exists, only individual interpretations of the individual interpretations of the world. ...learning mathematics world. ...learning mathematics should be thought of as a process should be thought of as a process of adapting to and organizing one’s of adapting to and organizing one’s quantitative world.quantitative world.

Learning is a social process in which Learning is a social process in which children grow into the intellectual life children grow into the intellectual life of those around them (Bruner, 1986). of those around them (Bruner, 1986). ...the constructivist classroom is ...the constructivist classroom is seen as a culture in which children seen as a culture in which children are involved not only in discovery are involved not only in discovery and invention but in social discourse and invention but in social discourse involving explanation, negotiation, involving explanation, negotiation, sharing, and evaluation.sharing, and evaluation.

When a teacher demands that When a teacher demands that students use set mathematical students use set mathematical methods, the sense-making methods, the sense-making activity...is seriously curtailed.activity...is seriously curtailed.

Conceptual and Conceptual and Procedural Procedural KnowledgeKnowledge

Instrumental and Instrumental and Relational Relational LearningLearning

Vygotsky’sVygotsky’sZone of Proximal Zone of Proximal

DevelopmentDevelopment

Previous LearningPrevious Learning

Self-DirectedSelf-DirectedLearningLearning

ZPDZPD

Things I have Things I have learned about learned about learning from learning from

Piaget and childrenPiaget and children

Piaget’s StagesPiaget’s Stages Sensori-motor (0-2)Sensori-motor (0-2)

[acquisition of language][acquisition of language]

Pre-operational (2-7) Pre-operational (2-7) [conservation (of number)][conservation (of number)]

Concrete operational (7-12)Concrete operational (7-12)[development of “adult” logic][development of “adult” logic]

Formal operational (12 up)Formal operational (12 up)

What Makes a Difference?What Makes a Difference?

1.1. Provide informal experiences Provide informal experiences (years) before formal instruction.(years) before formal instruction.

2.2. Find out what the student Find out what the student understandsunderstands (diagnosis). (diagnosis).

3.3. Plan instruction based upon what Plan instruction based upon what the student the student needsneeds..

4.4. Make it meaningful, develop the Make it meaningful, develop the concepts first.concepts first.

5.5. Make rich connections with previous Make rich connections with previous knowledge and experience. knowledge and experience.

6.6. Reasoning is more important than Reasoning is more important than answers!answers!

7.7. Use assessment to carefully Use assessment to carefully monitor learning.monitor learning.

8.8. Make cautious use of drill.Make cautious use of drill.

9.9. Increase instructional timeIncrease instructional time

10.10. Increase “time on task”Increase “time on task”

11.11.Start by making the child Start by making the child comfortable with new material.comfortable with new material.

12.12.Provide for success!Provide for success!

13.13.Help each child build confidence Help each child build confidence (self-validation).(self-validation).

14.14.Provide a sense of being valued.Provide a sense of being valued.

15.15.Attitudes of adults matter.Attitudes of adults matter.

16.16.Provide for recognition by peers.Provide for recognition by peers.

17.17.Make math fun some of the time.Make math fun some of the time.

18.18.Provide a challenge!Provide a challenge!

Intro to PRSIntro to PRS

Who is the future

Mrs. Dr. Robinson?

A

C

B

D

True or FalseTrue or False

Dr. Robinson is the world’s greatest Dr. Robinson is the world’s greatest professor…professor…

Conceptual Knowledge best Conceptual Knowledge best characterizes…characterizes…

A. Instrumental LearningA. Instrumental Learning

B. Relational LearningB. Relational Learning

Which is Which is NOTNOT a benefit of a benefit of relational learning?relational learning?

A. Extrinsic motivationA. Extrinsic motivation

B. Enhances memoryB. Enhances memory

C. Less to rememberC. Less to remember

D. It is self-generativeD. It is self-generative

Vgotsky’s Zone of Proximal Vgotsky’s Zone of Proximal Development…Development…

A. States that some students can not learn A. States that some students can not learn some concepts, while other, brighter some concepts, while other, brighter students, canstudents, can

B. Is a developmental theory which suggests B. Is a developmental theory which suggests correct timing of teacher interventions existscorrect timing of teacher interventions exists

C. Relates the idea of scaffolding concepts and C. Relates the idea of scaffolding concepts and knowledge to create more knowledge for a knowledge to create more knowledge for a studentstudent

D. Does not apply to mathematics educationD. Does not apply to mathematics education

True or FalseTrue or False

Constructivism means that every Constructivism means that every teacher creates his or her own teacher creates his or her own meaning and knowledge.meaning and knowledge.

Place these stages of Place these stages of Cogitative Thinking in Cogitative Thinking in Developmental OrderDevelopmental Order

A. Concrete operationalA. Concrete operational

B. Formal operationalB. Formal operational

C. PreoperationalC. Preoperational

D. Sensori-motorD. Sensori-motor

At which stage does a student At which stage does a student (child) display conservation?(child) display conservation?

A. Sensori-motorA. Sensori-motor

B. PreoperationalB. Preoperational

C. Concrete operationalC. Concrete operational

D. Formal operationalD. Formal operational