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• Author or Editor: Douglas H. Clements
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## Young Children's Ideas about Geometric Shapes

Atoddler, after some experimentation, puts a square peg into a square hole. What does she know about shapes? What more will she learn in preschool and elementary school. What might she learn?

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## Early Childhood Corner: The Earliest Geometry

In keeping with the early childhood chapter of Principles and Standards for School Mathematics, this department examines activities and children's thinking in geometry and, in the next issue, number. From prekindergarten to grade 12, the Geometry Standard addresses four main areas: properties of shapes, location and spatial relationships, transformations and symmetry, and visualization. For each area, we quote the goal of the Standard and the associated early-childhood expectations. We then present snippets of research and sample activities to develop ideas within each area with students.

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## Early Childhood Corner: Mathematics Everywhere, Every Time

“I'm first today!” “Then I want to be second. You gotta be third, Joon.”

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## Connecting Research to Teaching: Geometry and Proof

Educators continue to debate the relative emphasis that formal proof should play in high school geometry. Some argue that we should continue the traditional focus on axiomatic systems and proof Some believe that we should abandon proof for a less formal investigation of geometric ideas. Others believe that students should move gradually from an informal investigation of geometry to a more proof-oriented focus.

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## Using Logo Pseudoprimitives for Geometric Investigations

The computer screen in which the Logo turtle moves represents a small portion of a mathematical plane. A great deal of the geometry taught in junior and senior high school is essentially the study of such a plane and the objects that exist within it. Thus, creating and investigating shapes with the turtle clearly involves geometric thought. However, because many geometric ideas encountered in Logo are not covered in the standard curriculum, integrating them with textbook topics can be difficult. To allow traditional geometric topics to be used within the context of the exciting environment of Logo, a set of special procedures, called pseudoprimitives, has been created (Battista 1987). This article will describe how these procedures can be extended and used in junior and senior high school geometry.

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## Reviewing And Viewing

Reviewed here is a set of five drill-and-practice programs designed for preschool and elementary school children. The reviewer have evaluated these programs on this, basis—as situations for practice.

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## Learning of Geometric Concepts in a Logo Environment

To investigate the effects of computer programming in Logo on specific geometric conceptualizations of primary grade children, 48 third graders were randomly assigned to either a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo environment. The children were then interviewed to ascertain their conceptualizations of angles, shapes, and motions. In both groups children's notions of angle and angle measure were multifaceted and included a number of misconceptions, although performance was uniformly higher for the Logo group. The Logo children were more aware than the control children of the components of geometric shapes and were more likely to conceptualize geometric objects in terms of the actions or procedures used to construct them.

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## The Effects of Logo on Children's Conceptualizations of Angle and Polygons

It has been claimed that in Logo programming children learn mathematics by using concepts that aid them in understanding and directing the turtle's movements. The fundamental basis for this claim is that appropriate Logo environments can help children elaborate on, and become cognizant of, the mathematics implicit in certain kinds of intuitive thinking. To investigate this claim, 12 fourth graders were interviewed three times, at the beginning, middle, and end of 40 sessions of Logo graphics programming experience. The six Logo children, but not the comparison children, progressed from their original intuitive notions to more mathematically sophisticated and elaborate ideas of angle, angle size, and rotation. In addition, more Logo children explicitly mentioned geometric properties of shapes, indicating that they were beginning to think of the shapes in terms of their properties instead of as visual gestalts. Thus, there was support for the hypothesis that Logo experiences, especially those enriched with appropriate activities and discussions, can help children become cognizant of their mathematical intuitions and move to higher levels of geometric thinking.

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## Brief Report: Sex Differences in Rote-Counting Ability on Entry to First Grade: Some Observations

Currently, the issue of sex differences in mathematical abilities is of great concern. This article focuses on one early number skill: rote counting. Almost a century of research has produced equivocal results. These discrepancies might be accounted for by differences in methodology and data analysis. This report presents data on sex differences in rote-counting ability and illustrates how different data-gathering methods and different statistical treatments of the data can yield different results.

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