2. For science & mathematics what does it mean to KNOW/ UNDERSTAND/GRASP
The NRC understands there to be 4 strands of scientific proficiency:
The NRC understands there to be 5 strands of mathematical proficiency
know, use, and interpret scientific explanations of the natural world;
generate and evaluate scientific evidence and explanations;
understand the nature and development of scientific knowledge; and
participate productively in scientific practices and discourse.
conceptual understanding
procedural fluency
strategic competence
adaptive reasoning
productive disposition

3. Which One Does Not Belong? Why?4 8 16 36
Y=2^x
Y=x^2
Y=4x

4. Jean Piaget Stage Cognitive Skills Sensorimotor (infancy)
Uses senses to explore world. Egocentric perspective. Recognizes events may be caused by intentional actions. Realizes objects exist even when not present. Focuses on the present.
Preoperational
(toddler/early childhood)
Uses representational symbols. Rapidly develops language. Less dependence on senses. Classifies objects by single feature. Believes everything happens for a reason.
Concrete Operational
(elementary/adolescence)
Thinks logically about objects and events. Classifies objects by several features. Understands numbers. Realizes objects can appear in different forms.
Formal Operational
(adolescence/ adulthood)
Solves complex verbal and hypothetical problems. Thinks through symbols. Able to reason scientifically and logically. Capable of abstract thinking that includes past, present, and future.
(3:57)
According to Jean Piaget, intelligence develops in a necessary sequence of stages that are related to age. Piagets theories have given us profound insights into the cognitive development of children. This knowledge in turn, provides a framework for understanding how children think and for planning educational strategies. Using structured interviews with children from four to nine years old, Dr. David Elkind illustrates the development of transitive thinking and reversibility as children move from the pre-operational to concrete operational stage of cognitive development. Childrens construction of the unit concept, so basic to arithmetic and beginning reading, is illustrated in a well run first grade class. Dr. Elkind also examines some recent criticisms of Piagets theories. With David Elkind, Ph.D. This ia short clip from the film. Further information avaialble at

5. Jean Piaget
Piaget’s theories focused attention on the idea of developmentally appropriate education.
Teachers should provide learning environments, curriculum materials, and instruction that correlate with students’ stages of cognitive development.
They should introduce new concepts with realistic, hands-on activities followed by more abstract learning experiences.
Piaget’s developmental stages help teachers decide what to teach, how much to teach, and when to teach it.

6. Jean Piaget
Piaget also developed the schema theory in which students group similar types of information into schemata (categories) according to similarities and differences.
In view of this theory, effective teachers systematically plan introductory activities that connect new topics with students’ existing personal and academic experiences.

7. Jerome Bruner Three Sequential Modes of Learning:
Enactive: learning by doing
Iconic: learning by forming mental images
Symbolic: learning through a series of abstract symbols or representations

8. What Do They Have In Common?7 67 79 31

9. Constructivism
It echoes the basic premise of this ancient Chinese proverb: “I hear and I forget; I see and I remember; I do and I understand.”
In this student-centered approach, students are active participants in their own learning; teacher-centered strategies, such as lecturing, are minimized.
Students’ minds are no longer seen as blank slates or as empty vessels that need to be filled.
Constructivists believe that learners construct their own understanding and knowledge of the world based upon their experiences and reflection.

10. Constructivism
Students construct their own meaning by negotiating that meaning with others, making connections with and modifying prior conceptions, and addressing content in a variety of contexts.
Because students are active creators of their own knowledge, they are encouraged to ask questions, explore, and access what they know.
Students construct understanding of concepts themselves by connecting new learning with prior knowledge and experiences.

11. Constructivism
Instead of spending time memorizing material, filling in the blanks on worksheets, and repeating large numbers of similar problems, students learn to solve novel problems, integrate information, and create knowledge for themselves.
Teachers facilitate learning by building upon students’ preexisting knowledge, urging them to use active techniques (experiments, real-world problem solving) to create more knowledge, and then providing time for reflections upon what they have learned.

12. “Shaping Up” What did you learn?