The counting principles: counting: is the action of finding the number of elements os a finite set of objects. 1.The stable-order principle: to be able to count also means knowing that the list of words used must be un a repeatable order. 2. The once-one principle: this involves the assigning of one, and only one, distinct counting word to each of the items to be counted. 3. The cardinal principle: on condition that the one-one and stable-order principles have been followed, the number name allocated to the final objetc in a collectioin represents the number of items in that collection. 4. The order-irrelevance principle: this refers to the knowledge that the order in which items are counted is irrelevant. 5. The abstraction principle: the preceding principles can be applied to any collection of objects, whether tangible or not.
Piaget identifies 3 kinds of knowledge: Physical knowledge: it is the knowledge of real objects and properties that they can perceive. This knowledge is achieved by empirical abstraction, the child focuses on a certain property of objects, ignoring the others.
Social knowledge: these are names and conventions made up by people, is arbitrary and contextual only by being told or demostrated by other people, celebrated christmas on dec 25 for example.
Logical-mathematical knowledge: this is the creation of relationships. The brain buils neural connections which connect pieces of knowledge to one another to form new knowledge. Is constructed by each individual, inside his own head. It doesnt come from the outside. Is achieved by reflective abbstraction, and is a true construction of the mind rather than an observation of a certain property that already exists in objects.
Order, hierarchical inclusion: the child doesnt feel need to point objects pointing individually or mentally in a specific order,to ensure that he doesnt repeate or forget none. He doesnt need to arrange a spatial order the only important thing is the mental order of objects. Order inst the only ation performed with the objects. To cuantify the objects as a group, the child has to establish a relationship of hierarchical inclusion among them.
Van Hiele`s theory. 1. Visualization: students identity a shape for its appearance as a whole, but dont establish relationship between its elements. 2. Analysis: students begin to notice some properties of the figures, but arent able to observe the relationship between these properties. 3. Informal deduction: student can use informal logic to infer properties of the figures and can also recognise figures by its properties throught reasoning. 4. Formal deduction. 5. Rigor.