The researchers examined the use of video modeling by means of a handheld computer as an alternative instructional delivery system for learning basic geometry skills. Three high school students with learning disabilities participated in this study. Through video modeling, teacher-developed video clips showing step-by-step problem solving processes…

The purpose of this study was to explore word problems as a subject matter in mathematics textbook curricula. The motivation for the study derived from the following evidence: (a) American students find some word problems are more difficult than others (Garcia, Jimenez, & Hess, 2006; Riley & Green, 1988; Stern, 2001), and (b) one of the…

This article describes a mathematically rich problem about giant pencils, which encourages students to reason proportionally. (Contains 3 figures.)

This article relates how a class of elementary preservice teachers used technology to share their thinking about a topic that is traditionally difficult to model and discuss--dividing fractions. One type of technology they used, a digital camera, was familiar to most students, but the other type, a SMART Board[TM], was relatively unfamiliar.…

The purpose of this study was to investigate children's construction of 10s out of the 1s they have already constructed. It was found that, for many younger children, a dime was something different from 10 pennies even though they could say with confidence that a dime was worth 10 cents. As the children grew older, their performance improved.…

To deeply understand proportional reasoning, students must be able to differentiate proportional situations from nonproportional situations. Unfortunately, the emphasis on the use of proportional strategies in middle-grades mathematics has led students to develop a disposition to associate certain characteristics of problem formulation with the…

Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In "How We Think", esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the…

This study focussed on how secondary school students construct and use mathematical models as conceptual tools when solving word problems. The participants were 511 secondary-school students who were in the final year of compulsory education (15-16 years old). Four levels of the development of constructing and using mathematical models were…

This article presents a guided investigation into the spacial relationships between the centres of the squares in a Fibonacci tiling. It is essentially a lesson in number pattern, but includes work with surds, coordinate geometry, and some elementary use of complex numbers. The investigation could be presented to students in a number of ways…

Proportional thinking is the mathematical basis of a wide range of topics in the middle-school mathematics curriculum. While the concept is obvious in the traditionally-named ratio and proportion sections, proportional thinking is also the key to such diverse topics as rate, gradient of a linear function, similarity, trigonometry and percentage,…

Here, we investigate what loci are produced when a square of side-length one is allowed to rotate around a square of side-length n, where n is a whole number. We find that if i = 1, 2, 3 or 4 (mod 4), the loci obtained for n [congruent to] i (mod 4) all have the same symmetry and we show how the perimeter of each class can be determined. We also…

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers (Dixon, Deets, & Bangert, 2001). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation.…

A colleague was moving, and someone on the professional moving crew, upon hearing she was a mathematician, asked what happens when you repeatedly transfer water back and forth between two classes using a straw. The question is simple to solve if you alternate which glass you transfer from and to, but if more general patters are allowed, some…

Selectively retrieving a subset of previously studied information enhances memory for the retrieved information but causes forgetting of related, nonretrieved information. Such retrieval-induced forgetting (RIF) has often been attributed to inhibitory executive-control processes that supposedly suppress the nonretrieved items' memory…

In this paper we give a general solution to the problem of the damped harmonic oscillator under the influence of an arbitrary time-dependent external force. We employ simple methods accessible for beginners and useful for undergraduate students and professors in an introductory course of mechanics.