While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational…

Units coordination, defined by Steffe (1992) as the mental distribution of one composite unit (i.e., a unit of units) "over the elements of another composite unit" (p. 264) is a powerful tool for modeling students' mathematical thinking in the context of whole number and fractional reasoning. This paper proposes extending the idea of a…

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based…

Sampling distributions are fundamental for statistical inference, yet their abstract nature poses challenges for students. This research investigates the development of high school students' conceptions of sampling distribution through informal significance tests with the aid of digital technology. The study focuses on how technological tools…

Mathematics education often grapples with the challenge of teaching abstract mathematical concepts, particularly those existing in 3D space. Visualizing, manipulating, and comprehending these abstract objects can be a formidable task for learners. While 3D printing technology has found applications in various fields, its utilization in mathematics…

The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or "magnitudes" may be especially important for building robust…

While a significant amount of research has been devoted to exploring why university students struggle applying logic, limited work can be found on how students actually make sense of the notational and structural components used in association with logic. We adapt the theoretical framework of unitizing and reification, which have been effectively…

This paper addresses how secondary students might reason about amounts of change in covarying quantities. Two empirically based forms of covariational reasoning are distinguished. The first form-- reasoning about quantities as varying simultaneously and independently--supports tandem comparison of amounts of change. The second form--coordination…

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children's thinking, and learning situations…

In the authors' examination of various instructional programs, they observed that most provide all the necessary data to solve proportion problems, employ compatible numbers that are usually unrealistic, present numbers (data) in the order in which they are to be manipulated, discuss contexts that cannot be easily replicated, and present…

Since the game SET[R] was first introduced to the public in 1993, it has stimulated some interesting studies. While the game itself is rather straightforward, a plethora of decent mathematical questions lies beneath the surface. It is perhaps because the game ties in so closely with such an underlying mathematical term that its implications can be…

Discusses students' inability to make the connection between manipulative materials and pencil-and-paper calculations in mathematics instruction. Outlines the development of mathematical ideas through the concrete, representational, and abstract phases of instruction. An annotated bibliography listing teacher resources for representational-level…

Nondeterminism is an essential concept in mathematics and one of the important concepts in computer science. It is also among the most abstract ones. Thus, many students find it difficult to cope with. In this article, we describe some didactic considerations, which guided the development of a "Computational Models" course for high school…

Concepts from cognitive psychology are presented as an alternative framework to arithmetic instruction for students with disabilities. The approach, which emphasizes reasoning, communication, and problem solving, is applied to addition, division, multiplication, subtraction, and word problems. The purpose of arithmetic instruction is asserted to…

The development of an instructional model for teaching formal mathematical concepts (probability concepts) to disadvantaged high school students through computer programming and some results from a field test are described in this document. The instructional model takes into account both learner characteristics (cognitive, affective, and…