This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem. (Contains…

This paper examines three historical geometric constructions for handcrafting stringed instruments. Using elementary geometry--in particular, triangles--these methods can provide quite good rational approximations to the irrationals that arise from tuning instruments in equal temperament. Interestingly, continued fractions help explain the…

Four experiments were conducted in order to examine effects of notation--Arabic and verbal numbers--on relevant and irrelevant numerical processing. In Experiment 1, notation interacted with the numerical distance effect, and irrelevant physical size affected numerical processing (i.e., size congruity effect) for both notations but to a lesser…

When are the precursors of ordinal numerical knowledge first evident in infancy? Brannon (2002) argued that by 11 months of age, infants possess the ability to appreciate the greater than and less than relations between numerical values but that this ability experiences a sudden onset between 9 and 11 months of age. Here we present 5 experiments…

It has been argued that numbers are spatially organized along a "mental number line" that facilitates left-hand responses to small numbers, and right-hand responses to large numbers. We hypothesized that whenever the representations of visual and numerical space are concurrently activated, interactions can occur between them, before response…

This article reports challenges faced by prospective elementary teachers as they revisited whole number multiplication through a sequence of tasks that required them to develop and justify reasoning strategies for multiplication. Classroom episodes and student work are used both to illustrate these challenges, as well as to demonstrate growth over…

The connection between linear and 0-1 integer linear formulations has attracted the attention of many researchers. The main reason triggering this interest has been an availability of efficient computer programs for solving pure linear problems including the transportation problem. Also the optimality of linear problems is easily verifiable…

Place value is a phenomenon that has ominous implications for developing number sense and meaning and for using alternative algorithms and alternative representations within whole number arithmetic. For the most part, school programs examine place value at a surface level, with a primary focus on having the student identify or state a number value…

A k-dimensional integer point is called visible if the line segment joining the point and the origin contains no proper integer points. This note proposes an explicit formula that represents the number of visible points on the two-dimensional [1,N]x[1,N] integer domain. Simulations and theoretical work are presented. (Contains 5 figures and 2…

All the familiar numbers (rationals, algebraic numbers and a few transcendental numbers) have measure zero. (MM)

As education continues to progress schools are constantly seeking innovative ways to cultivate and enhance achievement for all students. As a result many public schools are pushing toward the inclusion model. This model includes co-taught instruction to meet the many needs of special education students. This research study was implemented to…

Purpose: Elementary teachers' understanding of mathematics is a significant contributor to student success with mathematics. Consequently, teacher educators are frequently charged with the responsibility of supporting the development of prospective elementary teachers' mathematics content knowledge as they re-learn concepts in ways they are…

The purpose of the research is to explore second graders' concept of number development and quantitative reasoning. For this purpose, there were two stages of trials for the children. The first trial was concrete objects. After three months, the children participated in the second trial of half concrete objects. Since understanding the process of…

Proof is a central component of mathematicians' work, used for verification, explanation, discovery, and communication. Unfortunately, high school students' experiences with proof are often limited to verifying mathematical statements or relationships that are already known to be true. As a result, students often fail to grasp the true nature of…