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• Author or Editor: Constance Kamii
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## One Point of View: Arithmetic: Children's Thinking or Their Writing of Correct Answers?

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## Measurement of Length: How Can We Teach It Better?

Measurement of length is taught repeatedly starting in kindergarten and continuing in grades 1, 2, and beyond. However, during the past twenty-five years, according to the National Assessment of Educational Progress (NAEP), the outcome of this instruction has been disappointing. What is so hard about measurement of length? This article explains, on the basis of research, why instruction has been ineffective and suggests a better approach to teaching. Teachers can use these practical applications for teaching measurement in the classroom.

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## Multiplication Games: How We Made and Used Them

Several multiplication games that were used to develop third graders' fluency with multiplication facts.

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## Multiplication with Fractions: A Piagetian, Constructivist Approach

On the basis of piaget's (1954, 1960) Constructivism, Kamii (1989, 1994) has demonstrated that children in the primary grades can invent their ownprocedures for solving multidigit problems with whole numbers. A significant finding of this research is that when children are not taught algorithms, such as those of “carrying” and “borrowing,” their number sense and knowledge of place value are far superior to those of students who have been taught these rules. Warrington (1997) extended this work to the fifth- and sixth-grade level and described an approach to “teaching” division with fractions without teaching the algorithm of “invert and multiply.” This article describes a constructivist approach to multiplication with fractions.

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## Achievement Tests in Primary Mathematics: Perpetuating Lower-Order Thinking

The Curriculum and Evaluation Standards (NCTM 1989) states that if we want to improve the nation's mathematics education, it is necessary to change the current method of evaluation that depends on standardized achievement tests. The National Research Council (1989) is even more explicit about the harmful effects of achievement testing.

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## Teaching Place Value and Double-Column Addition

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## Research Into Practice: Constructivism and First-Grade Arithmetic

For arithmetic instruction in the first grade, we advocate the use of games and situations in daily living in contrast to the traditional use of textbooks, workbooks, and worksheets. Our position is supported by the research and theory of Jean Piaget, called constructivism, as well as by classroom research (Kamii 1985, 1990).

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## Identification of Multiplicative Thinking in Children in Grades 1–5

Textbooks present multiplication as merely a faster way of doing repeated addition. However, research has shown that multiplication requires higher-order multiplicative thinking, which the child develops out of addition. Three hundred thirty-six children in grades 1–5 were interviewed individually using a Piagetian task to study their development from additive to multiplicative thinking. Multiplicative thinking was found to appear early (45% of second graders demonstrated some multiplicative thinking) and to develop slowly (only 48% of fifth graders demonstrated consistently solid multiplicative thinking). It was concluded that the introduction of multiplication in second grade is appropriate but that educators must not expect all children to use multiplication, even in fifth grade.

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## Brief Report: The Older of Two Trees: Young Children's Development of Operational Time

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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## Elapsed Time: Why Is It So Difficult to Teach?

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.