1. Jean Piaget
Piaget believed that people go through definite developmental stages in their lives. Each stage must be completed before a person can attain the next stage.

2. Piaget’s Stages of Cognitive Development
Sensorimotor (toddler to 2 years old)
Preoperational (2 to 6 years old)
Concrete Operational (6 to 12 years old)
Formal Operational (over 12 years old)
Piaget’s approach of asking questions and building on the answers of children aligns well with the need for oral discourse as seen in the NCTM Principles and Standards of School Mathematics.

3. Piaget’s Stages continued…
Sensorimotor
“Out of sight, out of mind”
Actions on Objects: Kids will touch, taste, squeeze, etc.
Preoperational
Know objects exist, can describe
Actions on Reality: Children will fill containers from others, may hide containers within others, etc.
Concrete Operational
Can reason with concrete objects: Children will figure out which containers hold the most, least, equal amounts.
Actions on Operations
Formal Operational
Can reason without concrete objects (with former experience)
Operations on Operations

4. Lev Vygotsky
People continually construct and restructure knowledge. No two people have the same understanding of concepts. Teacher intervention is important!

5. Vygotsky’s Constructivism
Zone of proximal development
When people can produce & analyze a complex skill on their own.
Teacher questioning techniques
Children can be guided to better mathematical understandings as they progressively analyze complex skills on their own. Children can internalize actions more quickly with the guidance of good analyzing questions from his/her teacher.