The article investigates the patterns that arise in the convergence of numerical methods, particularly those in the errors involved in successive iterations, using data analysis and curve fitting methods. In particular, the results obtained are used to convey a deeper level of understanding of the concepts of linear, quadratic, and cubic…

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as…

Research has shown that educated children and adults have access to two ways of representing numerical information: an approximate number system (ANS) that is present from birth and allows for quick approximations of numbers of objects encountered in one's environment, and an exact number system (ENS) that is acquired through experience and…

Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their…

Growing awareness of the importance of education in the early years has led professional organizations and policy makers to consider the effectiveness of mathematics education for young children. Factors such as educational environments, early years curricula and teacher education are some of the many aspects that have been examined. This paper…

Even though basketry is an age old occupation in Ghana and the world over, it appears mathematics tasks and activities involving the designs and structures have remained unnoticed and inadequately tapped for early school instructions. This qualitative survey therefore, purposively sampled four teachers, two basket artisans and six pupils in the…

Historically there has always been a lively research discussion on whether the development of number concept should be considered as innate, or whether the catalyst for the development of the number concept -- and for counting -- is hearing number words combined with seeing concrete examples. One can recognize these theories as the nativist view…

Learning through picture books is a method which is widely used in preschools to teach mathematics. However, the research findings on the benefits of using picture books in learning mathematics remain inconclusive. This study aimed to i) investigate the effects of a picture book entitled "What Are the Numbers?" (a picture book made up…

Middle-grades mathematics prospective teachers (PSTs) participated in three iterations of a design experiment that explored their current conceptualizations of integer and binomial multiplication; this study analyzed whether that knowledge changed after engaging in a series of scaffolded tasks while using area models as concrete manipulatives.…

I present a viable learning trajectory for prospective elementary teachers' number sense development with a focus on whole-number place value, addition, and subtraction. I document a chronology of classroom mathematical practices in a Number and Operations course. The findings provide insights into prospective elementary teachers' number sense…

While previous studies mainly focused on children's additive and multiplicative reasoning abilities, we studied third to sixth graders' "preference" for additive or multiplicative relations. This was investigated by means of schematic problems that were "open" to both types of relations, namely arrow schemes containing three…

The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention was centered on the number line and incorporated key principles from the science of learning. Sixth graders (N = 51) who struggled with fraction concepts were randomly assigned at the student level to the experimental intervention (n = 28) or…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…

These are the appendixes for the report, "What Grade 7 Foundational Knowledge and Skills Are Associated with Missouri Students' Algebra I Achievement in Grade 8?" To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…