Research on dimensional comparison processes has shown that comparing one's own verbal achievements with poorer results in math leads to a higher self-concept of verbal ability. It has been argued that when observers make inferences about a student's academic self-concept, they do not usually have access to dimensional information. In the present…

The fraction 10000/9801 has an intriguing decimal expansion, namely 1.02030405... In this paper, we investigate the properties of this fraction via an arithmetical approach. The approach also yields a class of fractions whose decimal expansions involve higher-dimensional analogues of the integers, the n-dimensional pyramidal numbers, thereby…

We argue that to test preschoolers' understanding of counting, one has to use tasks that relate counting to the goal of doing arithmetic, as counting and arithmetic principles are mutually constrained. A naturalistic study in the preschool classroom led to the development of an ''arithmetic-counting'' task, where counting was being related to the…

A method for generating sums of series based on simple differential operators is presented, together with a number of worked examples with interesting properties.

This paper focuses on the representation of a proper fraction "a"/"b" by a decimal number base "n" where "n" is any integer greater than 1. The scope is narrowed to look at only fractions where "a","b" are positive integers with "a" less than "b" and "b" not equal to 0 nor equal to 1. Some relationships were found between "b" and "n", which…

This article discusses divisibility tests for any prime number.

The paper details a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers…

This article describes the impact of thinking about addition and subtraction facts instead of rotely memorizing them.

Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…

Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure. This guide provides eight specific recommendations intended to help teachers, principals, and school administrators use Response to Intervention (RtI) to identify students who need…

Every student should understand and use all concepts and skills from the previous grade levels. The standard is designed so that new learning builds on preceding skills. Communications, Problem-solving, Reasoning & Proof, Connections, and Representation are the process standards that are embedded throughout the teaching and learning of all…

Every student should understand and use all concepts and skills from the previous grade levels. The standard is designed so that new learning builds on preceding skills. Communications, Problem-solving, Reasoning & Proof, Connections, and Representation are the process standards that are embedded throughout the teaching and learning of all…

Aim: Subcortical band heterotopia (SBH) or "double cortex" is a malformation of cortical development resulting from impaired neuronal migration. So far, research has focused on the neurological, neuroimaging, and genetic correlates of SBH. More recently, clinical reports and small sample studies have documented neuropsychological dysfunction in…

This article proposes an instructional idea where students can figure out an individual's secret personal information using the power of mathematics, particularly the power of algebraic thinking. The proposed examples in this article start with a personalized context that other people do not know and end up with generalized patterns of solutions.…

Background: Understanding and solving problems involves different levels of representation. On the one hand, there are logico-mathematical representations, or problem models (PMs), which contain information such as "the size of the flock changed from 31 sheep to 42" while, on the other hand, there are more qualitative representations, or…