1. EEX 4251 Developing Mathematical Knowledge Dr. Matthew C. Robinson, Spring 2008

3. The Math Lesson

4. 3 4 11 12 13 + 1 + 2 + 1 + 2 + 4 3 4 11 12 13 + 1 + 2 + 1 + 2 + 4 4111214102 101 101 113 1001 100 + 10+ 4+ 14+ 201+110 101 101 113 1001 100 + 10+ 4+ 14+ 201+110 111110100212021000 The Lesson

5. 101 1004 203 1002 1013 +202+ 103+114+ 113+ 101 101 1004 203 1002 1013 +202+ 103+114+ 113+ 101 104 112 13 112 104 +201+202+101+ 104+203 104 112 13 112 104 +201+202+101+ 104+203 Practice 1013 1112 1112 1200 1114 1013 1112 1112 1200 1114 1100 1104 11410111102 1100 1104 11410111102

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

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

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

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

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

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

12. Conceptual and Procedural Knowledge

13. Instrumental and Relational Learning

14. Life’s “Great Lessons” “Almost all, who have ever fully understood arithmetic, have been obliged to learn it over again in their own way.” Warren Colburn, 1849

15. Ways of Thinking Enactive - Manipulate objects Iconic - Think about (visualize) objects Symbolic - Meaningful manipulation of symbols (as short- hand)

16. Vygotsky’s Zone of Proximal Development Previous Learning Self-DirectedLearning ZPD

17. Things I have learned about learning from Piaget and children

18. Piaget’s Stages Sensori-motor (0-2) Sensori-motor (0-2) [acquisition of language] Pre-operational (2-7) Pre-operational (2-7) [conservation (of number)] Concrete operational (7-12) Concrete operational (7-12) [development of “adult” logic] Formal operational (12 up) Formal operational (12 up)

20. What Makes a Difference? 1.Provide informal experiences (years) before formal instruction. 2.Find out what the student understands (diagnosis). 3.Plan instruction based upon what the student needs. 4.Make it meaningful, develop the concepts first.

21. 5.Make rich connections with previous knowledge and experience. 6.Reasoning is more important than answers! 7.Use assessment to carefully monitor learning. 8.Make cautious use of drill. 9.Increase instructional time 10.Increase “time on task”

22. 11.Start by making the child comfortable with new material. 12.Provide for success! 13.Help each child build confidence (self- validation). 14.Provide a sense of being valued. 15.Attitudes of adults matter. 16.Provide for recognition by peers. 17.Make math fun some of the time. 18.Provide a challenge!

25. Intro to PRS

26. Who is the future Mrs. Dr. Robinson? A C B D

27. True or False Dr. Robinson is the world’s greatest professor… Dr. Robinson is the world’s greatest professor…

28. Conceptual Knowledge best characterizes… A. Instrumental Learning B. Relational Learning

29. Which is NOT a benefit of relational learning? A. Extrinsic motivation B. Enhances memory C. Less to remember D. It is self-generative

30. Vgotsky’s Zone of Proximal Development… A. States that some students can not learn some concepts, while other, brighter students, can B. Is a developmental theory which suggests correct timing of teacher interventions exists C. Relates the idea of scaffolding concepts and knowledge to create more knowledge for a student D. Does not apply to mathematics education

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

32. Place these stages of Cogitative Thinking in Developmental Order A. Concrete operational B. Formal operational C. Preoperational D. Sensori-motor

33. At which stage does a student (child) display conservation? A. Sensori-motor B. Preoperational C. Concrete operational D. Formal operational