Simple examples are really encouraging in the understanding of rearrangements of infinite series, since many texts and teachers provide only existence theorems. In the absence of examples, an existence theorem is just a statement and lends little confidence to understanding. Iterated sums of double series seem to have a similar spirit of…

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when…

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay…

The method for geometric transforms (probability generating functions) is used to study the expected number of observations until the pattern 123123 . . . 123 is obtained. These results provide a first generalization of similar problems considered by other authors. (Contains 1 figure.)

The Law of Sines and The Law of Cosines are of paramount importance in the field of trigonometry because these two theorems establish relationships satisfied by the three sides and the three angles of any triangle. In this article, the authors use these two laws to discover a host of other trigonometric relationships that exist within any…

In this article, the author takes up the special trinomial (1 + x + x[squared])[superscript n] and shows that the coefficients of its expansion are entries of a Pascal-like triangle. He also shows how to calculate these entries recursively and explicitly. This article could be used in the classroom for enrichment. (Contains 1 table.)

The study focused on the development of the concept of fractions in a group of 11- and 12-year-old students who were deaf or hard of hearing. The approach implemented in the study relied extensively on the use of games with very little formal instruction. It emphasized the development of appropriate language to facilitate an understanding of the…

Although some bright students in primary school are able to organise numerical data into classes, most attend to the characteristics of individuals rather than the group, and "see the trees rather than the forest". How can teachers in upper primary and early high school teach students to organise large sets of data with widely varying…

This article presents several activities (some involving graphing calculators) designed to guide students to discover several interesting properties of Fibonacci numbers. Then, we explore interesting connections between Fibonacci numbers and matrices; using this connection and induction we prove divisibility properties of Fibonacci numbers.

Child-staff ratios are an important quality indicator. They are often collected by observing one randomly selected classroom several times during a 2-h period on a single day. However, it is unclear whether these measures represent the ratios that children actually experience during most of their time in care. This study compared ratio data…

It is over 25 years since Wood, Bruner and Ross (1976, "Journal of Child Psychology and Psychiatry," 17, 89-100) introduced the idea of "scaffolding" to represent the way children's learning can be supported. Despite problems, this metaphor has enduring attraction in the way it emphasises the intent to support a sound foundation with increasing…

This article describes surprising misconceptions revealed by a fifth-grade student during a series of interviews about probability.

When teachers learn to take time to understand how students are thinking about mathematics, they will be better able to develop students' mathematical understanding. Teachers must become investigators--of mathematics and of student thinking--in their own classrooms. When teachers explore mathematics, analyze student thinking, and write cases from…

The problem of solving mathematical equations can be quite tough for some students hence they face a great difficulty when applying ideas to the actual process. Students in algebra classes are taught coding in which they write down what they will need to do to solve the equation and this coding makes the students more adept at solving equations…

A conceptually sound, research-based method is presented to make sense of students' thinking about length. It is suggested that the levels of framework is an important tool for improving instruction and formative assessment as well as effectively diagnosing and remediating students' difficulties in learning about length. Some of the measurement…