A class of 14 kindergartners in Japan was videotaped while playing a card game in groups of three involving the placement of cards in numerical order. The children were followed up in first grade, and it was found that development in one area of logico-mathematical knowledge (for example, the making of temporal relationships) stimulates development in other areas (such as classification and numerical reasoning). This specific, interrelated mode of structuring was not expected and suggested that it is unwise for adults to try to plan a sequence of development. The implication of the findings is that it may be better to provide "natural" play activities that encourage children to think logico-mathematically than to conceptualize specific standards for 3-to-6-year-olds' mathematics education.

Four examples of physical-knowledge activities are described and analyzed on the basis of Piaget's theory. These are playful activities like Pick-Up Sticks in which children act on objects mentally and physically to produce a desired effect.The objective of physical-knowledge activities is to develop children's logico-mathematical knowledge. Therefore, it is not the activities themselves that are important. What is important is the thinking children do while they play because it is by thinking that children construct logico-mathematical knowledge, and logico-mathematical knowledge serves as the framework for children to construct all knowledge.Data are presented about the achievement in mathematics of two groups of low-SES first graders who came to school without any number concepts. One group was given physical-knowledge activities during the math hour for half a year instead of math lessons. The other group received traditional math instruction throughout the year. The first group did better in mental arithmetic at the end of the school year, demonstrating the importance of a solid logico-mathematical foundation.

One hundred students, 25 each in grades 2, 4, 6, and 8, were individually interviewed. A cylindrical beaker with five equidistant levels marked on it was presented first, and a pear-shaped separation funnel was also presented. The funnel had a “faucet” at the bottom that allowed water to start and stop running into the beaker. The interviewer asked the child to mark the water levels on the funnel when the water had run to the first level in the beaker, to the second level, to the third level, and so on. The levels on the funnel thus varied a great deal depending on its diameter. The interviewer then asked, “Did the water take the same amount of time to go down from this level to the next in the funnel as it did to go up from this level to the next in the beaker?” It was found that correct judgments about durations were not made before eighth grade (the criterion being 75% of the eighth graders giving correct answers). This grade level was considerably later than the age of 7 or 8 (second grade) that Piaget reported.

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.

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 I I 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.

The purpose of this study was to investigate children's construction of 10s out of the Is they have already constructed. Ninety-eight children in Grades K-4 were individually interviewed in a store game in which they were the storekeepers and the interviewer was the customer buying candy. We tried to find out how children gave change when payment was made with a dime or a dime and a few pennies for purchases up to 9 cents. It was found that, for many younger children, a dime was something different from 10 pennies even though they could say with confidence that a dime was worth 10 cents. As the children grew older, their performance improved. Educational implications are discussed.

Among the first graders who came to our Title-I school one year, we found twenty-six children who had no understanding of number concepts. In the assessment at the beginning of the school year, these children could not conserve number with eight counters. They could count out four chips, but when we hid some of the chips and asked, “How many am I hiding?” the children gave random answers, such as, “Ten.” Our challenge was that we were required by law to teach an hour of arithmetic to these children every day despite the fact that they had not yet developed an understanding of number concepts.

Do your students like to play tic-tac-toe? If so, do you think of games as time fillers or part of your educational program? In our classrooms, we use this game as a serious educational activity. The purpose of this article is to explain why we value this and similar games and how we use them. We base our opinion on research conducted in Japan (Nagahiro, Kato, and Miyakawa 2003) and the United States (DeVries and Fernie 1990; Kamii 2000; Kamii and DeVries 1996), and our observations in many classrooms.

To find out if children could make functions before age 4, 73 children aged 1 to 4 were encouraged to imitate the use of a lever to make a beanbag fly up. Functions are mental relationships that preoperational children can make between 2 things at a time in a unidirectional way (Piaget, Grize, Szeminska, & Bang, 1968 Piaget, J., Grize, J.-B., Szeminska, A. and Bang, V. B. 1977. Epistemology and psychology of functions, Boston: Dreidel. (Original work published 1968)[Crossref] , [Google Scholar]/1977). The child's construction of the following 3 functions was hypothesized and confirmed: (a) As a function of being pushed down, the up end of the board (the lever) goes down; (b) as a function of this descent, the down end of the board goes up; and (c) as a function of this ascent of the board, the beanbag flies up. Three developmental levels were found, and educational implications are discussed.