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• Author or Editor: Daiyo Sawada
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## Mathematics as Reasoning—Episodes from Japan

As recommended in the NSTM's curriculum and evaluation Standards for School Mathematics (1989, 29), the study of mathematics should emphasize reasoning so that students can—

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## Brief Reports: Multisensory Information Matching Ability and Mathematics Learning

The multisensory approach to learning has found wide acceptance in elementary school mathematics based, in part, on the writings of Bruner, Piaget, and Dienes. Basic knowledge about what kind of sensory experiences to provide for which pupils in what sequence has been the focus of considerable inquiry. A study by Sawada and Jarman (1978) identified and supported the usefulness of a particular paradigm for assessing sensory integration ability within and across sense modalities. The study reported here is the second of a series of studies using this paradigm.

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## Piaget and pedagogy: fundamental relationships

Over the past ten years, and with increasing frequency, the name of Jean Piaget has become a familiar sight to the eyes of mathematics teachers, especially at the elementary school level. Numerous books have appeared (Almy 1966; Athey and Rubadeau 1970; Beard 1969; Copeland 1970; Furth 1970), along with a large number of journal publications, th at address themselves to the topic of Piagetian implications for teaching.

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## Symmetry and Tessellations from Rotational Transformations on Transparencies

The title of this article may be a bit ostentatious, but it describes a simple yet rather dramatic procedure for producing symmetrical tessellated patterns. I feel that mathematics is all around us if we would only care to look. It is easy to remind students of mosaic tiling patterns, hexagonal or other interlocking paving patterns, or Escher's work. Recently I discovered that simple dot grids in a square (fig. I) or in a triangle (fig. 2) can produce tessellations (interlocking patterns that cover a surface).

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## Operations from Number Sequences

The idea of a number sequence is rarely regarded as a basic concept in elementary school mathematics. When we do study number sequences we do so (1) indirectly, as when we speak of the “whole number sequence,” wherein the idea of a sequence appears only in a manner of speaking—we really are much more interested in the whole numbers per se: or (2) as a topic in enrichment, as when we study the triangle number sequence (1, 3, 6, 10, …) or some other sequence from number theory. But even in the enrichment setting, we are more interested in the numbers themselves than in the fact that they form a sequence.

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## Computer Power in Primary Grades: Mathematics with Big Trak™

All is quiet in the house. Even the cat, snuggled on his cushion beside the refrigerator, seems to be asleep. Suddenly, from the hallway, a vehicle emerges. chugging along quietly. It halts for a moment, a if getting its bearing, turn slightly to the left, traverses the island in the kitchen, turns right toward the refrigerator, and finally stop in front of the cat. Three flashes of light, accompanied by “space beeps,“ are launched to-ward a surprised cat. Laughter is heard in the hallway as the vehicle backs up and retrace its journey down the hallway (see fig. 1).

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## Mathematical Symbols: Insight through Invention

Over the years, like a pendulum. the emphasis in mathematics education has swung from a focus on concepts and understanding (e.g., the new-math movement) on the one hand to skill with facts and algorithms (e.g., the back-to-basic movement) on the other. Currently. children can adequately perform algorithms, but they may do so with little understanding of the underlying concepts (Resnick 1982, 136–55). In part, the difficulty lies in students having lo t sight of the role of symbols in mathematical thinking. The development of approache. that help children integrate the insight of symbolic understanding with the power of algorithmic technique should be of value. Accordingly, the intent of this article is to suggest how children can be guided to see and personally feel the power and simplicity that thinking with and about mathematical symbols can bring to their algorithmic competence. Although, for the sake of concretene and pecificity, attention hall be confined to computation, stress shall be placed on an approach that the reader may find generalize to other areas.

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## Study 3-1: An Assessment of a Selected Set of Piagetian-Derived Operators for the Generation of Effective Learning Hierarchies

When Klein's study (reported in this journal) is viewed so that the forest can be seen as well as the trees, one realizes that the problems of sequencing can be analyzed as two types: (a) problems of transfer within a cluster (intracluster transfer) and (b) problems of transfer between clusters (intercluster transfer). Klein limited his study to an examination of intracluster transfer by focusing on the ramifications of the MR matrix as applied to a given objective, such application giving rise to a cluster of objectives. The problem of intercluster transfer (going from one cluster to another) was not attended to by Klein.

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## Magic squares: extensions into mathematics

Many a potentially dull practice session on addition and subtraction has been transformed into a lively encounter with mathematics through the introduction of magic squares. Figure 1 shows three of the more familiar ways in which magic squares are used. Normally work with magic squares in the elementary school does not go beyond the types of examples shown there. Further exploration of magic squares is usually limited to finding simple procedures for constructing 3 × 3 squares, 4 × 4 squares, and n × n squares where n is odd. The procedures are frequently presented in a rather rote fashion since the emphasis is on practice for addition, not on the properties of magic squares per se.

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## NCTM's Standards in Japanese Elementary Schools

In recent years, the NCTM's Standards (1989, 1991) and Asian mathematics education (Becker et al. 1990; Stevenson and Stigler 1992; Stedman 1994; and many others) have, each in its own right, received a great deal of attention. I believe, however, that to look at the connections between the two areas would greatly benefit teaching. In this article, five classroom situations taken from observational studies of mathematics teaching in Japanese elementary schools are described and interpreted from the perspective of the two Standards documents (1989, 1991). More specifically, the classroom situations are examined from the perspective ot the first four standards found in the Curriculum anil Evaluation Standards (1989): Mathematics as Problem Solving. Mathematics as Communication. Mathematics as Reasoning, and Mathematical Connections.