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Sidney G. Hacker

Edited by Howard Eves

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.

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Sam Rhodes and Jessica Duggan

Cryptography-based investigations can help students develop conceptual understandings.

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Ivan D. Stones

In the search for nonroutine material for a class on the history of mathematics, I became interested in the patterns that can be discovered within number triangles. “Looking for patterns” is a mathematical activity that not only is challenging and fun for students but also helps develop problem-solving techniques.

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Margaret A. Farrell

Some good suggestions on how a competency-based teacher-education program might focus on nontrivial tasks and measure them in ways that are not merely based on checklists.

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Faye B. Clark and Constance Kamii

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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Carol Ann Alspaugh

This study was designed to determine what student characteristics seem to influence proficiency in symbolic and algebraic computer programming. 50 students in a beginning university computer programming course were administered the Thurstone Temperament Schedule, IBM Programmer Aptitude Test, and the Watson-Glaser Critical Thinking Appraisal. These test scores, along with SCAT scores and a measure of the students' mathematical background, were correlated with measures of student programming proficiency. Those students who possessed a mathematical background of approximately 2 college calculus classes, who scored low on “impulsiveness” and “sociability,” and who scored high on “reflectiveness,” appeared to have “aptitude” for computer programming.

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Katherine E. Lewis and Marie B. Fisher

Although approximately 5–8% of students have a mathematical learning disability (MLD), researchers have yet to develop a consensus operational definition. To examine how MLD has been identified and what mathematics topics have been explored, we conducted a systematic review of 164 studies on MLD published between 1974 and 2013. Findings indicate that (a) there was great variability in the classification methods used, (b) studies rarely reported demographic differences between the MLD and typically achieving groups, and (c) studies overwhelmingly focused on elementary–aged students engaged in basic arithmetic calculation. To move the field toward a more precise and shared definition of MLD, we argue for standards for methodology and reporting, and we identify a need for research addressing more complex mathematics.

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Lynn M. McGarvey

A child's decision-making photo activity about pattern identification presents implications for teaching and learning patterns in the early years.

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Mary L. Wheeler

Identification codes have been used for many years. We've seen a large increase in the use of scanners and computers in the last two decades. As a result, the need for identification codes increased tremendously. These codes are an assignment of numbers to an item for identification of information about that item. They are very useful with computers. information can be stored in a condensed format, and with the use of algebraic algorithms the accuracy of that information can be checked by using what is called a check-digit scheme. Identification codes are also very useful for secwity information. The use of a simple or complicated mathematical algorithm allows the information to be disguised.

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Alan Barson and Lois Barson

Because this issue concerns early childhood education, IDEAS for this month consists of activities for the primary grades. The grids offer practice with patterns, number and shape identification, and computation.