National Council of Teachers of Mathematics

This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.

On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

Atmospheric and Oceanic Fluid Dynamics

Although it is evident in routine decision-making and a crucial vehicle of rationalization, commensuration as a general social process has been given little consideration by sociologists. This article defines commensuration as the comparison of different entities according to a common metric, notes commensuration's long history as an instrument of social thought, analyzes commensuration as a mode of power, and discusses the cognitive and political stakes inherent in calling something incommensurable. We provide a framework for future empirical study of commensuration and demonstrate how this analytic focus can inform established fields of sociological inquiry.

/pdf/commensuration-as-a-social-process-1wnmqj0u8f.pdf

Commensuration as a social process

■ Abstract It is widely but incorrectly believed that the t-test and linear regression are valid only for Normally distributed outcomes. The t-test and linear regression compare the mean of an outcome variable for different subjects. While these are valid even in very small samples if the outcome variable is Normally distributed, their major usefulness comes from the fact that in large samples they are valid for any distribution. We demonstrate this validity by simulation in extremely non-Normal data. We discuss situations in which in other methods such as the Wilcoxon rank sum test and ordinal logistic regression (proportional odds model) have been recommended, and conclude that the t-test and linear regression often provide a convenient and practical alternative. The major limitation on the t-test and linear regression for inference about associations is not a distributional one, but whether detecting and estimating a difference in the mean of the outcome answers the scientific question at hand.

David Eugene Smith

Papers

History of Mathematics

A source book in mathematics

Capitalism and Arithmetic: The New Math of the 15th Century.

A History of Mathematics in America Before 1900

The Casting-Counter and the Counting-Board