In this workshop, we will continue to reflect on a models and modeling perspective to understand how students and teachers learn and reason about real life situations encountered in a mathematics and science classroom. We will discuss the idea of a model as a conceptual system that is expressed by using external representational media, and that is…

An approach is presented for determining unidimensional scale estimates that are relatively insensitive to limited inconsistencies in paired comparisons data. The solution procedure--a minimum-cost network-flow problem--is presented in conjunction with a sensitivity diagnostic that assesses the influence of a single pairwise comparison on…

This article describes several methods from discrete mathematics used to simulate and solve an interesting problem occurring at a holiday gift exchange. What is the probability that two people will select each other's names in a random drawing, and how does this result vary with the total number of participants? (Contains 5 figures.)

We investigate short-term learning from hints and feedback in a Web-based physics tutoring system. Both the skill of students and the difficulty and discrimination of items were determined by applying item response theory (IRT) to the first answers of students who are working on for-credit homework items in an introductory Newtonian physics…

This article discusses divisibility tests for any prime number.

A model for population growth is described. (SD)

In this paper the mathematical structures of Bourbaki, the Piagetian developmental structures, and Polya's heuristic structures are compared. (SD)

The approximate nature of measurement is stressed, with examples given of elementary school mathematics problems which confuse reality with its mathematical model. Examples relating measurement to mathematical structure are given. (DT)

Discusses a real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the bounce data of a ping-pong ball. Describes the lesson in which students collect bad data then fill in the missing parabolas that result using critical components of parabolas and…

This article analyses the modeling approach used by one student in a business problem. It is argued that if we use previous frameworks we are not able to classify the students' approach to modeling as purely theoretical or empirical. Instead the student used a theoretical approach when constructing a real model, but abandoned it when she had to…

This theme issue features five articles profiling Indiana University faculty whose work on various campuses continues to broaden and advance knowledge about cognitive science. The articles in the journal are: "A Matter of Time" (Karen Grooms) which discusses the work of Robert F. Port; "Perceiving as a Complex System" (Tom…

This document consists of three papers. The first, "A Parallel Model of (Sequential) Problem Solving," describes a parallel model designed to solve a class of relatively simple problems from elementary physics and discusses implications for models of problem-solving in general. It is shown that one of the most salient features of problem…

Presents mathematical games that involve a problem-solving matrix, multicolored cubes, and three-dimensional dominoes. The work of Alexander MacMahon is highlighted. (MA)

Describes some unsolved problems in geometry, as well as some recently solved ones. Indicates that each advance generates more problems than it solves, thus ensuring a constant growth in unsolved problems. (GA)

Presents examples of direct modeling and invented strategies of students for problem solving. Teachers are encouraged to capitalize on children's abilities to use these invented strategies and in time guide the student toward a standard, more efficient strategy. (AIM)