We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.

This study investigates two sixth grade students' dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students' formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically…

The current study examines the technical adequacy of four Early Numeracy Curriculum-Based Measurement (EN-CBM) screening tasks: "Oral Counting" (OC), "Number Identification" (NI), "Quantity Discrimination" (QD), and "Missing Number" (MN). Results from 59 kindergarten students assessed in the fall and spring reveal moderate to high test-retest and…

This study examined the neuronal correlates of reading Roman numerals and the changes that occur with extensive practice. Subjects were scanned by functional Magnetic Resonance Imaging (fMRI) three times the first day of the experiment and once following two to three months of practice. This allowed comparison of brain activations with varying…

This paper reports selected findings from a study of number sense proficiency of students aged 13 to 16 years in a state in Malaysia. A total of 1756 students, from thirteen schools in a state in Malaysia participated in this study. A majority (74.9%) of these students obtained an A grade for their respective year-end school examinations. The…

Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is…

Bases such as 5 and 12 provide the same structural place value benefits as base 10. However, when numbers less than one are concerned, base 10 provides friendly decimals for the most common fractions of half, quarter, three-quarters. Base 5 is not user friendly at all in this regard. Base 12 would provide nice dozenimals(?) for the same…

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of "numbers", extending the rational number system, adequate for measuring continuous quantities. Moreover, that such "numbers" are in one-to-one correspondence with points on a "number line". But…

Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…

This classroom note includes the following sections: Introduction; The General Case; and Related Formulas.

Authors use combinatorical analysis to prove some interesting facts about the Fibonacci sequence.

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…

Although many studies have examined the alignment of state standards with large-scale assessment and instruction, fewer have attended to alignment concerning alternate assessments for students with significant disabilities. This study was designed to (1) compare expectations in one state's alternate assessment (AA) with curricular priorities…

Simplicity in number naming in a language (e.g. "ten-two" in Chinese is simpler than the irregular "twelve" in English) has been used to explain cross-cultural disparities in children's computational competence. In contrast to previous research focusing only on whether children can provide the correct answers, in this study (N =117 and 92) we…

In the renewed "Primary Framework for Mathematics" for England, great emphasis is given to calculation and its prerequisites (DfES, 2006). Expectations are increased for calculations and the recall of number facts, with mental calculation owning a higher profile, while progression in written calculation is clarified. The greater focus on…