We show that the total distance traveled by an under-damped oscillating spring mass system with sinusoidal displacement results in a nice, closed-form expression.

Mathematical argumentation is generally thought to be the paradigm of cogent reasoning. The concept of mathematical proof thus seems to be associated with necessity and enforcement, but not with freedom; however, in various ways a reference to freedom is also needed to understand the phenomenon of mathematical proof, for example, to distinguish it…

In spite of the widely accepted need for language-responsive subject-matter teaching, few teachers are prepared for this challenge due to the lack of empirically founded subject-specific professional development (PD) programs for language-responsive classrooms. The design research study presented in this article pursues the dual aim of (a)…

In this study, I examined seven first-year linear algebra students' linear independence schemas. Data came from participants' interview responses to a set of nine questions. The analysis focused on the identification of concepts and connections pertaining to plans and activations. Overall, the findings revealed the existence of routinized plans,…

Learning is a complex phenomenon. Analyzing its development can provide knowledge about regularities and differences in students' progress, which is needed to better understand learning. The aim of this study is to examine the development of students' Linear Algebra Schema through an introductory one semester course. For that purpose, Action…

Arguably the largest obstacle freshmen face in their mathematics courses is their unfamiliarity with the language of mathematics. Addressing this problem right at the start seems like a sensible strategy, as comprehension of mathematical communication helps students in all mathematics courses they will take. In this paper we discuss general…

This paper reviews the research on equitable mathematics teaching practices that focus on providing fair opportunities to learn for all children. The studies conducted with in-service teachers are examined in descriptive, integrated and critical ways to explore the differences and similarities across the body of research. Three divergent themes…

In this article we are analyzing which spherical symmetry groups can be found and which ones cannot among the eggs with wax-resist patterns most frequently used in the Carpathian Basin. We are also trying to answer the question why exactly these symmetry decorations can be found. Before that, we are providing a review on the history of decorating…

The development of multiple solutions for a given problem is important for learning mathematics. In the present intervention study, we analyzed whether prompting students to construct multiple solutions (more precisely: prompting them to apply multiple mathematical procedures to real-world problems) and prior self-efficacy influenced students'…

This paper sets forth a construct that describes how many undergraduate students understand mathematical terms to refer to mathematical objects, namely that they only refer to those objects that satisfy the term. I call this students' pronominal sense of reference (PSR) because it means they treat terms as pronouns that point to objects, like…

The idea of using fingers as a key component in arithmetic development has received a great deal of support, much of which is based on neuroscientific evidence. However, this body of work pays limited attention to how fingers are used and possible different outcomes in arithmetic problem solving. The aim of our paper, based on an analysis of 126…

Counting is an important functional skill that is embedded throughout people's daily lives, which can result in numerous meaningful opportunities through which students can learn. Counting, especially early counting, provides a foundation for the development of later mathematics understanding and is highly predictive of later mathematics…

Research suggests that teachers struggle to find effective ways to introduce proof. In 1940, in an article in this journal, Roland Smith argued that being aware of student misconceptions in geometry is the first step in preparing to address the fundamental challenges of learning to prove. Through careful study, he identified and analyzed…

Two trends play an increasing role in shaping the long-term prospects of children across the United States: a growing STEM (science, technology, engineering, and mathematics)-based economy and rising income segregation. Together, they suggest that the future well-being of U.S. children may be based increasingly on the contemporary interplay of…

Fragile understanding is where new learning begins. Students' understanding of new concepts is often shaky at first, when they have only had limited experiences with or single viewpoints on an idea. This is not inherently bad. Despite teachers' best efforts, students' tenuous grasp of mathematics concepts often falters with time or when presented…