Optimizing the dimensions of a soda can is a classic problem that is frequently posed to freshman calculus students. However, if we only minimize the surface area subject to a fixed volume, the result is a can with a square edge-on profile, and this differs significantly from actual cans. By considering a more realistic model for the can that…

The staircase and fractional part functions are basic examples of real functions. They can be applied in several parts of mathematics, such as analysis, number theory, formulas for primes, and so on; in computer programming, the floor and ceiling functions are provided by a significant number of programming languages--they have some basic uses in…

Trigonometry is one of the fundamental topics taught in high school and university curricula, but it is considered as one of the most challenging subjects for teaching and learning. In the current study Mason's theory of attention has been used to examine undergraduate student's perception of the transformation of sinusoidal functions. Two types…

Mathematics is seen by many people as the best way to get a good life and a good career. It is also thought as an assistant to understand life and the world and to produce ideas about them. Therefore, new reform studies are being held to construct a new system that assists students to learn mathematics in a comprehensive way (Dursun & Dede,…

In two studies (N's = 55 and 54), we examined a basic form of conceptual understanding of rational number arithmetic, the direction of effect of decimal arithmetic operations, at a level of detail useful for informing instruction. Middle school students were presented tasks examining knowledge of the direction of effects (e.g., "True or…

The "Making Meaning for Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The twenty-nine cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation of specific…

In this exploratory study we considered how one teacher's understanding of proportional reasoning related to his teaching. We relied on Knowledge in Pieces as a lens for considering the teacher's knowledge organization as a way to make sense of his understanding. We did so by collecting data from two instruments and using Epistemic Network…

The study investigated the effects of Solve-Reflect-Pose Strategy (SRP) on pre-service mathematics teachers' algebraic knowledge for teaching in Nigeria. A pre-test-post-test quasi experimental design was employed. Intact classes were used and in all, 182 pre-service mathematics teachers' participated in the study (92 in the experimental group…

Pivot Professional Learning, in consultation with the Australian Association of Mathematics Teachers, summarises their study into the characteristics of Australian mathematics teachers. The study reinforced that Australia has many excellent teachers of mathematics, but can improve by sharing excellent teaching practices of its expert teachers…

This paper explores how a group of pre-service elementary school teachers training to become mathematics teachers for elementary schools arrived at generalizations based on patterns. Two representative problems were investigated with these preservice teachers. The focus of this study was how these preservice teachers analyze and symbolize…

The purpose of this study was to investigate secondary mathematics teachers' conceptions of mathematical literacy, which are important to consider when is essential to address in designing and implementing effective approaches. Data sources included semi-structured interviews with 16 in-service mathematics teachers from different types of…

This study investigated the accuracy of three fraction measures (i.e., fraction number line estimation accuracy, general fraction concepts, and fraction arithmetic) for screening fourth graders who might be at risk for mathematics difficulties. Receiver operating characteristic (ROC) curve analyses assessed diagnostic accuracy of the fraction…

Learning progressions (LPs) describe the development of domain-specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity…

The first author's initial attempts at a whole-class mathematics discussion were woefully dysfunctional. Ten second graders showing their answers to a missing addend question about the number of children left riding a school bus did not result in the mathematical epiphany that the textbook promised. She did not give up on using mathematical…

Early math instruction is the subject of increasing attention by early childhood researchers. There is evidence to suggest that early math skills are highly predictive of later academic achievement but that there are disparities in early math skills among young children. The purpose of this study was to examine the effectiveness of a systematic…