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Constance Kamii and Kelly A. Russell

Piaget (1971) made a distinction between intuitive (preoperational) time and operational (logico-mathematical) time. According to Piaget, operational time develops around 7–8 years of age and is characterized by children's ability to deduce, for example, that if A was born before B, A will always be older than B. When time is still intuitive, children base their judgments of age on what is observable (e.g., people's height). With the aid of 11 pictures of an apple tree and a pear tree taken on 6 consecutive birthdays, 184 children in grades K–5 were individually asked, at a specific time, if two trees were the same age or if one was older than the other. Operational time was demonstrated by 79% of these children by grade 3.

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Constance Kamii and Kelly A. Russell

Based on Piaget's theory of logico-mathematical knowledge, 126 students in grades 2–5 were asked 6 questions about elapsed time. The main reason found for difficulty with elapsed time is children's inability to coordinate hierarchical units (hours and minutes). For example, many students answered that the duration between 8:30 and 11:00 was 3 hours 30 minutes (because from 8:00 to 11:00 is 3 hours, and 30 more minutes is 3 hours 30 minutes). Coordination was found to begin among logicomathematically advanced students, through reflective (constructive) abstraction from within. The educational implications drawn are that students must be encouraged to think about durations in daily living and to do their own thinking rather than being taught procedures for producing correct answers to elapsed-time questions.