Kamii, Constance, and Leslie Baker Housman. _Young Children Reinvent Arithmetic: Implications of Piaget’s Theory._ New York: Teachers College Press, 2000. # Progressive Summary # Key Points ## Chapter One - CK asked educators how children acquire the concept, not the word, of a number such as 8. She got various responses such as: "they learn it from experience", "they count objects", or "they make a one-to-one correspondence". - She compares mathematics education to folk medicine, because it is operating without a scientific explanation of what causes mathematical ideas. In contrast, physicians treat illnesses by identifying a cause, or if they can't, then they treat the symptoms. - Piaget was more of an epistemologist than a psychologist. He was interested in how we acquire knowledge. He believed it was important to study how this occurred in the early stages of man's evolution, but since he had no access to this, he studied children as proxies for ancestral humans. - Piaget belongs to an epistemological tradition that goes back to the debates between the empiricists (Hume, Locke) and rationalists (Kant, Descartes, Spinoza). The empiricists believed that the mind was a blank slate, and that we get all our knowledge from the senses. The rationalists believed that our knowledge of things such as cause and effect, and mathematical relationships, could not possibly be had from the senses, so they must have an innate source. - Piaget divided knowledge into 3 kinds: physical, social and logico-mathematical. Physical and social knowledge we get partially from the outside world. But logico-mathematical knowlege, which is knowledge of relationships, is constructed from innate abilities interacting with the environement. This is called constructivism. - Numbers up to 4 or 5 are perceptual numbers, because you can tell the difference between them just by looking at them. But it's not easy to tell at a glance the difference between 7 and 8 objects. - He tested children at different ages to see how their ability to see relationships progressed. Relationships such as sameness, difference, size. One key difference he found between 4 and 8 year olds is that the former could not see parts and wholes at the same time, whereas the latter could. For instance, if shown 2 cats and 6 dogs, and asked if there were more dogs than animals, the 4 year old would say yes, because they could not think of the parts (cats and dogs) at the same time as the whole (animals). But the 8 year olds were able to do this. Piaget felt that this proved that logico-mathematical relationships were innate, because the 4 year olds could _see_ the parts and wholes, but they were lacking the mental representation. ## Chapter Two - Piaget distinguished between symbols and signs. Symbols resembled the objects to be counted, and children could draw them on them on their own. Signs were more abstract, and were the product of convention. Examples are mathematical notations, musical notations, and Morse code – all require social transmission. # Resonances # Quotes -