Volume 95, Number 3, March 1995

Table of Contents

Abstracts

Table of Contents

Abstracts

• Expanding Students' Conceptions of the Arithmetic Mean

Ruth Ann Meyer, Christine Browning, & Dwayne Channell
Western Michigan University 

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        The arithmetic mean is the most commonly used measure of central tendancy; nevertheless, many students who can add all the elements of a data set and then divide that sum by the number of elements do not truly understand the concept of mean. This article presents four activities designed to help elementary and middle school students develop a concept of mean. To bring about a desirable level of understanding, all computational formulae and algorithms in mathematics should be preceded by experience emphasizing conceptual understanding. Since that is not the normal instructional sequence for the arithmetic mean, the activities presented in the article assume previous exposure to the computational algorithm for the arithmetic mean.
 
• A Comparison Between Mathematics Textbook Content and a Statewide Mathematics Proficiency Test

Donald G. Chandler and Patricia A. Brosnan
The Ohio State University 

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        Seven mathematics textbook series used for Grades 1 through 8 were examined. Textbook pages were counted as arithmetic, measurement, geometry, data analysis, or algebra. Percentages of mathematics textbook content were compared with percentages of mathematics content on the Ohio Ninth Grade Proficiency Test. Areas of greatest mismatch were arithmetic, measurement, and algebra. The overall percentage of arithmetic in the textbook series was about twice that on the test; the percentage of measurement was less than half that on the test; and the percentage of algebra was a third or less than that on the test. Results are also presented by grade level, and implications for teaching are discussed.
 
• Use of Inappropriate and Inaccurate Conceptual Knowledge to Solve an Osmosis Problem

June Trop Zuckerman
State University of New York at New Paltz 

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        Two outstanding high school science students each generated a correct answer to an osmosis problem. The solution processes are noteworthy because the problem solvers did not blindly use algorithms. They relied, instead, on conceptual knowledge which was inaccurate and inappropriate to the problem. They thought the osmosis problem was about air pressure, and some of their knowledge about air pressure was inaccurate. Thus, even when students rely on conceptual knowledge to solve a problem, correct answers need not indicate adequate understanding. Characteristics of the problem solvers, salient properties of the problem that could contribute to the problem misrepresentation, and spurious correct answers are identified. Finally, some instructional recommendations and research questions are presented.
 
• Interpreting Negative Results in an Angle Maximization Problem

David R. Duncan and Bonnie H. Litwiller
University of Northern Iowa 

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        The correct interpretation of positive and negative results in a situation involving mathematical modeling poses problems for some students. This article presents a situation in which differential calculus is used with inverse trigonometric tangent functions to maximize an angle measure. A negative distance measure ultimately results, thus requiring a reconsideration of assumptions inherent in the initial figure.
 
• The Number 71.428571 in Hindu Cosmology

Enoch Haga
Pleasanton Unified School District (Retired) 

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        Zimmer, in his 1962 book, Myths and Symbols in indian Art and Civilization, speculates that the number 14, used to divide a kalpa into manvantaras, was chosen in an attempt to reconcile a mahayuga chronology with a tradition of flood heroes. Each manvantara was divided by 71 and an unspecified fraction to produce the mahayuga. The purpose of this research was to find this fraction and discover the reason for its use. The repeating cyclic fraction, 1/7, provides a plausible explanation for use of the number 71.428571 (71 and 3/7) in cosmology; for 22/7 as a value for (symbol); and for concepts of time and circle division related to breaths per minute. The discovery of cyclic decimal fractions by Vedic scholars likely contributed to cyclic theories of cosmology and specifically to the concept of infinity in mathematics.
 
• Implementing Problem-Based Learning in Science Classrooms

Shelagh A. Gallagher
Chicago Academy of Science
William J. Stepien
Social Science Research Institute
Beverly T. Sher
College of William and Mary
David Workman
Illinois Mathematics and Science Academy 

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        Many new curricular and instructional models must be developed or adapted as the nation moves towards educational reform in science classrooms. This article describes how problem-based learning, and innovative curricular and instructional model developed in medical graduate school programs, has been adapted for use in elementary and high school settings. Included in the integration of problem- based learning and science are components of all problem-based episodes including initiating learning with an ill-structured problem, using the problem to structure the learning agenda, and teacher as metacognitive coach, with important goals of a reformed science curriculum such as learning based on concepts of significance, student-designed experiments, and development of scientific reasoning skills.
 
• Dynamic Polygons and the Graphing Calculator

Joseph A. DiCarluccie
Ossining High School 

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        A match stick puzzle is used as a springboard for a secondary classroom investigation into mathematical modeling techniques with the graphing calculator. Geometric models, through their corresponding area formulas, are constructed, tested, and analyzed graphically to fit specified problem conditions. These specialized versions are then synthesized into generalizations that determine a polygon containing integral sides with a given perimeter and area.