that five pennies spread out on a table is the same number as five pennies pushed close together.
Until they reach that understanding -- which is a developmental step, not something taught -- other number concepts are a waste of time.
I would like to think they're only calling kindergarten work "algebra" and "geometry" to impress adults -- but they're doing so many other stupid things in school these days. So who knows?
http://educationnext.org/much-too-early/
Those who believe in academic training for very young children make a fundamental error: They fail to recognize that there are different levels of understanding in math and reading. Learning to identify numbers and letters is far different from learning to perform mathematical operations and to read with understanding. This is easy to support. “Sesame Street” has run for more than 30 years. Children today know their numbers and letters earlier than ever before. Many know them by age two. Yet children today are not learning math or reading any earlier or better than did children before there was “Sesame Street.” Learning the names of numbers and letters is only the first step in the attainment of true numerical understanding and reading comprehension.
Take the concept of numbers. The three levels of numerical understanding—nominal, ordinal, and interval—correspond to different forms of scaling. Nominal numbering is the use of a number as a name, such as the numbers basketball players wear on their uniforms. By the age of two or three, children can use numbers in the nominal sense. By the age of four or five, children can begin to use ordinal numbers; they can order things according to quantitative differences. For instance, they can arrange a series of size-graded blocks or sticks from the smallest to the largest. Once the arrangement is complete, however, they are not able to insert a new, intermediate-sized element into the perceptual array.
It is only at age six or seven, when they have attained what Piaget calls “concrete operations,” that children can construct the concept of a “unit,” the basis for understanding the idea of interval numbers. To attain the unit concept, children must come to understand that every number is both like every other number, in the sense that it is a number, and at the same time different in its order of enumeration. Once children attain the unit concept, their notion of number is abstract and divorced from particular things, unlike nominal and ordinal numbers. Mathematical operations like addition, subtraction, and multiplication can be performed only on numbers that represent units that can be manipulated without reference to particular things.
The interval concept of numbers is an intellectual construction. It builds on children’s practice in classifying things (attending to their sameness) and in seriating them (attending to their difference). At a certain point, and with the aid of concrete operations, children are able to bring these two concepts, of sameness and difference, together into the higher-order concept of a unit, which brings together the ideas of sameness and difference. It is only when children understand that something can be the same and different that they have a true understanding of quantity. Learning the names of numbers and rote counting are less important in this attainment than is practice in classifying and seriating many different materials.
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