The purpose of the study was to investigate the effects of the instruction with mathematical modeling on pre-service mathematics teachers' mathematical modeling performance. The participants were 24 pre-service elementary mathematics teachers. A mixed method approach was used to conduct the research. Each week, the participants were given two…

This article examines how visual representations may mediate the teaching and learning of mathematics over time in Japanese elementary classrooms. Using the Zone of Proximal Development Mathematical Learning Model (Murata & Fuson, 2006; Fuson & Murata, 2007), the process of mediation is explicated. The tape diagram, a central visual…

Discusses a classroom-teaching example provided in this issue. Comments that teachers, using curricular materials and activities, adapt, elaborate on, or invent the context of learning in ways that ensure that students study important central concepts in science and mathematics, and that the understandings students develop are accurate and rich.…

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)

Proposes a system to help at-risk students develop problem solving skills. Provides some teaching methods and evaluation ideas. (KHR)

Models for division are discussed: counting, repeated subtraction, inverse of multiplication, sets, number line, balance beam, arrays, and cross product of sets. Expressing the remainder using various models is then presented, followed by comments on why all the models should be taught. (MNS)

This is the fourth and concluding article in a series on using mathematical techniques from history to provide an interesting way for middle graders to practice math. (Editor)

An analysis of methods of teaching children to solve verbal arithmetic problems is presented together with transcriptions of interviews in which children solve problems by reference to problem types. (SD)

A model for division of fractions using money as manipulative material is presented. Eight levels are described, ranging from the development of language and concept introduction through types of problems to rule discovery and application. (MNS)

Suggestions for teaching problem solving which include the use of general models, variation of illustrations and terminology, and diagrams are illustrated for comparison multiplication word problems. (MN)

An approach is illustrated that introduces selected models of addition and subtraction to students who have previously learned a model for either operation. (JT)

A model that can be effectively used to develop the notion of function and provide varied practice by using "real world" examples and concrete objects is covered. The use of Popsicle-sticks is featured, with some suggestions for tasks involving functions with one operation, two operations, and inverse operations covered. (MP)

A manipulative model for the addition, subtraction, and multiplication of integers is described in which a carpet square with a hole in it and the plug that fits in the hole are "opposites" of each other. Several other models for integer multiplication are briefly described. (MN)

Described is the development of a simulation, or model of an existing congested pedestrian crossing situation by elementary school students in order to conduct trials of their solutions. (MN)

An alternative explanation is presented for the pattern of relative difficulty of arithmetical word problems to that suggested in a 1983 article. It is argued, that for many problems, much of the symbolic manipulation is accomplished externally, not internally. (MNS)