This article investigates the numbers [image omitted], originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C[subscript n], we develop some divisibility properties for C[subscript n]. In particular, we establish that [image omitted], where f[subscript k]…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

The availability of sophisticated computer programs such as "Wolfram Alpha" has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is…

Early elementary school students are expected to solve twelve distinct types of word problems. A math researcher and two teachers pose a structure for thinking about one problem type that has not been studied as closely as the other eleven. In this article, the authors share some of their discoveries with regard to the variety of…

Two 12-question estimation instruments were designed to compare the differences of estimating strategies used by the 8th-graders when solving contextual and numerical problems. Both instruments are parallel, meaning that the numbers used in both instruments are the same; however, they were presented differently. One hundred and ninety-eight…

Research on toddlers' mathematical knowledge is sparse. Studies on children's mathematical competencies before school age have mostly focused on older children. Few of the previous studies have included large groups of toddlers, few have been conducted in natural settings, and few have been directed at a broad field of mathematical knowledge. The…

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research…

A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key…

Reflecting concerns about student attainment and participation in mathematics and science, the Effecting Principled Improvement in STEM Education ("epiSTEMe") project attempted to support pedagogical advancement in these two disciplines. Using principles identified as effective in the research literature (and combining these in a novel…

As students contribute to social and sociomathematical norms, they will often reorganize their own understandings. As such, the purpose of this article is to describe the ways in which prospective elementary teachers contributed to the norms that were established and "re"-established throughout an entire semester in an undergraduate…

This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999... and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999..., but this definition depends upon accepting properties of the real…

In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and…

The current study examined the development of two effects that have been found in single-digit multiplication errors: relatedness and distance. Typically achieving (TA) second, fourth, and sixth graders and adults, and sixth and eighth graders with a mathematics learning disability (MLD) performed a verification task. Relatedness was defined by a…

Although students' invented strategies typically prove to be effective in the improvement of students' mathematical understanding, little is known about how preservice teachers interpret and respond to student-invented strategies on whole number multiplication. This study investigated the nature of 25 preservice teachers' interpretations of and…

This article addresses an important problem of graphing quantitative data: should one include zero on the scale showing magnitude? Based on a real time series example, the problem is discussed and some recommendations are proposed.