1. What are the capabilities of children at different age levels? What can children do? What do they start with? What are the processes by which they achieve their competence?
2. Piaget: Primary principle of development is adaptation to the environment. Accommodation: modify oneself to fit the environment; and assimilation: change things in the environment so that they can be used.

Stages of development--sensory-motor, preoperational thought (egocentrism, irreversibility, centering), concrete operations (conservation tasks), formal operations (symbol manipulation); but decalage is very important, can find examples of the early stages at all ages with new materials or domains.
3. Actual complexity in Piagetian type problems. E.g. Conservation of Number. Try to think of how you could get a computer to solve the task. Setting up an algorithm requires understanding the task. Each group of objects is a unit. How do you know this? Data seems to show that children seem to have an innate concept of a pairing method to identify equal cardinality, but they need to learn how to order and pair the members. When a child is asked to count a set of objects she is likely to err both on the sequence of numbers and the ability to know which are counted and which has yet to be counted.
4. In order to solve some problems children may have to learn to ignore some obvious information, perceptual size is ignored in favor of historical continuity (conservation of volume); habitual location is ignored in favor of most recent location (object permanence), etc. Some conservation tasks can be thought of as gaining perceptual and cognitive structuring (learning what goes together, and how different components are related) followed by the application of particular algorithms to the newly organized system (conservation of number, balance beam).
5. More complex problems: Each of these requires more complex algorithms to be solved. Partial algorithms lead to predictable erroneous solutions. Mixed juice problems as on pp. 306-7. Noelting (1980), Case (1978) mix juice and water. Keep adding dimensions needed for solving successive problems. Ages 3.5 to about 10. No clear stages as they move from low level to mathematical solutions. Different dimensions. Number, blending principle, quantity.
6. Siegler’s approach is the same. (Balance beam) More dimensions of the problem can be noticed. Much older children, ages 7 to 17. Here they had to be able to do basic mechanics--principle of torque, Number, distance, side, relation between number and force of gravity, Simple machines and mechanical advantage, all had to be considered.
7. Vygotsky: Zone of Proximal Development emphasizes cultural differences and the role of teaching, scaffolding supplied by expert, social precedes individual, complex skills have to be shaped. Luria and cross-cultural differences in cognition show cultural influences.
8. Evidence of sophisticated cognitive skills in very young children: E.g. Identify objects, continuity of objects in contour and through space, movement of objects. Learning common nouns. Naming dolls and boxes.
9. Chi: With the right kind of experience and training, children can do very sophisticated categorization. Bruner: Children develop the skill to incorporate more information in a single view so that they can "notice" disparities among groups.

We gradually develop competence in different domains. One real problem is finding the component processes that are necessary to do any of the tasks. This can include perceptual integration; object conceptualization, pragmatic uses, causal analyses, physical dynamics, abstract relations, understanding number, appropriate knowledge of the elements, learning the relations between the component parts and how to maneuver between them. One of the very interesting ways to see how many component processes are involved is to try to write an algorithm to do the task. Generate a set of efficient procedures to get there!