A relative extrema optimization problem is one in which the domain of the objective function (i.e. the function whose maximum or minimum value is to be found) is an open interval. An absolute extrema optimization problem is one in which the domain of the objective function is a closed interval. Analysis of task-based interviews conducted with 12…

In this study we investigated the development of 245 kindergarteners' mathematics problem solving (MPS) at two time points within 7 months using a cognitive diagnostic test, which measured three cognitive components (mathematical knowledge and skills, semantic understanding, and quantitative reasoning) with eleven cognitive attributes. The effects…

At the start of every school year, teachers typically wonder what their new students know coming into the grade level. Sometimes, though, they place too much emphasis on what children do not know or have forgotten over summer break. After all, those gaps can be glaring. Helping their students grow is like helping an acorn grow; they must watch for…

In "Principles to Actions: Ensuring Mathematical Success for All," NCTM (2014) defines productive struggle as students delving "more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions" (p. 48). Hiebert and Grouws (2007, p. 387)…

The prime objective of this research was to investigate whether the Montessori method of learning helped kindergarten pupils improve their mathematical proficiency, critical thinking and problem-solving skills, besides training them to be responsible learners. Quantitative, qualitative, and observational methods were employed in the investigation.…

Algebra is critical to high school graduation and college success, but student achievement in algebra frequently falls significantly below expected proficiency levels. While existing research emphasizes the importance of quality algebra instruction, there is little research about how to conduct problem analysis for struggling secondary students.…

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…

The purpose of this study is to contribute insights into how collaborative activity can help promote students' mathematical understanding. A group of six high school students (15- to 16-year olds) worked together on a challenging probability task as part of a larger, after-school, longitudinal study on students' development of mathematical ideas…

This paper reports on a mixed-methods study of 111 Advanced Placement calculus students' self-reports of their graphing calculator use, comfort, and rationale for trusting a solution produced with or without a graphing calculator when checking written work. It was found that there was no association between gender, teacher-reported mathematical…

Superitem test based on the SOLO model (Structure of the Observing Learning Outcome) has become a powerful alternative assessment tool for monitoring the growth of students' cognitive ability in solving mathematics problems. This article focused on developing a superitem test to assess students' algebraic solving ability through interview method.…

Several decades ago, V. A. Krutetskii conducted a multiyear study to investigate the various types of thinking that academically advanced, or as he called them, gifted mathematicians used. Following an in-depth look at Krutetskii's nine ways of thinking, a model is proposed that will provide direction for teachers in selecting problems. The model…

A solution to the Rubik's Cube was introduced to an eighth grade mathematics class. The purpose of this study was to determine if an introduction to a solution to the Rubik's Cube could enhance students' problem-solving abilities, increase their general interest in mathematics, and enhance students' problem solving self-efficacy. Pre and Post…

This study offers an analysis of the learning of practicing teachers as they acquire a deeper knowledge of mathematics. While some professional developers have shifted part of their focus to helping practicing teachers acquire a deeper knowledge of mathematics (e.g., Stein & Silver, 1996), the results from studies often describe what translates…

For the gifted mathematics student, early mastery of concepts and skills in the mathematics curriculum usually results in getting more of the same work and/or moving through the curriculum at a faster pace. Testing, grades, and pacing overshadow the essential role of creativity involved in doing mathematics. Talent development requires creative…

This article discusses the structuring of mathematics to challenge the mathematically gifted elementary student. Theoretical background examines the nature of the learning task and the role of the teacher. An example is given of three related tasks that scaffold the learning experience and help the student extend his/her mathematical thinking…