What Children Know About Numerals Before Being Formally Taught and Immediately Afterward
Infants are able to match the number of drumbeats to the number of objects in a visual display, rats can learn how many times they need to press a lever in order to receive areward, and a11 known human cultures have developed words for counting. Yet, mathematics is generally considered one of the most difficult subjects at school, and mathematical i11iteracy is a worldwide phenomenon (Paulos, 1995). What happens to our natural propensity for mathematics? Researchers speak of insufficient mathematical education, psychological blockage, and fa11acious, romantic ideas about mathematics. Psychologists and many mathematics teachers believe there is a problem because there is a divorce between the mathematics involved in day-to-day life and the symbolic representations used when teaching at school. Classroom instruction in mathematics is likened
According to Greeno, notations are viewed positively as a useful aid in a particular place or in a particular process, as when cookingj they are indeed consid-
ered helpful in communicating ideas. But, the advice is not to confound them with the real thingj notations are a tool, but "conceptual environments" are the "actual territory." Apparently, difficulties with mathematics stem from the ill-defined relation between these two entities: the notational and the conceptual. But to solve these difficulties, attention should be dedicated to the conceptual territoryj leaming stems from experiencing the conceptual territory. The idea is clear, if someone wants to leam about a particular pI ace (i.e., its routes, its rules, and its available resources), then the best thing to do is to get direct experience rather than resorting to a map. This is the particular view about the relationbetween domains ofknowledge and notation that was characterized in the introduction as the "derivative view." According to this view, knowledge of a domain, in this case knowledge of mathematical concepts, paves the way for understanding the notation of that particular domain.