This paper examines prospective mathematics teachers' knowledge of asymptotes and asymptotic behaviour of functions in calculus. They are university students and future facilitators of knowledge in upper secondary education. We constructed a "reference epistemological model" to describe the knowledge about asymptotes for upper secondary…

Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…

This article aims to delve into the meaning of school mathematical concepts by their semantic analysis. Concretely, this paper aimed primarily to describe and characterize the meaning of derivative of a function at a point expressed by mathematics pre-service teachers. In order to achieve this goal, we use a semantic framework, in which the…

The study discussed here aims to describe students' understandings of the definition of a mathematical function, which was achieved through a pilot case study of clinical interviews with four participants -- two ninth graders and two twelfth graders. The participants were recruited from the same urban public high school in the northeast of the…

The relation to operands (RO) principles describe the relation between operands and answers in arithmetic problems (e.g., the sum is always larger than its positive addends). Despite being a fundamental property of arithmetic, its empirical relation with arithmetic/algebraic problem solving has seldom been investigated. The current longitudinal…

This study diagnoses and explores year eight students' misconceptions and mistakes in algebra. An in-depth exploration of students' misconceptions were carried out by conducting interviews based on the students' work as outlined in a cognitive map. Based on the cognitive maps, it can be seen that the students emphasize procedural knowledge rather…

This action research study examined the effects of professional development on teaching for conceptual understanding in Algebra I. "Conceptual knowledge refers to knowledge of the underlying structure of mathematics-the relationships and interconnections of ideas that explain and give meaning to mathematical procedures" (Eisenhart et…

This study was performed to determine preservice elementary mathematics teachers' solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers'…

Introductory group theory provides the foundational course on algebraic structures. Yet, we know little about students' underlying conceptual understandings. In this paper, I introduce the Group Theory Concept Assessment (GTCA), a measure created for the purpose of conducting large-scale studies of student conceptual understanding in group theory.…

Current recommendations for school mathematics include teaching algebra as a K-12 content stranding beginning in the elementary-school grades. The focus of this study was on preservice elementary mathematics teachers' knowledge of algebra content in the intended curriculum for K-8 mathematics. In the Common Core State Standards for Mathematics…

This study concerns preservice teachers' knowledge of student thinking at the secondary level. In particular, it examines in what ways a three-week unit integrating content and student thinking was beneficial in developing preservice teachers' knowledge of student thinking associated with big ideas in algebra--such as variation, Cartesian…

This study explores learning vector addition and demonstrating the understanding of vector addition. Vector addition is a key skill in mathematics and science education, but there is a limited amount of research that focuses on this topic. My study investigates how an instructional design framework "concreteness fading" can be used for…

The purpose of this study was to investigate how the 8th graders interpreted the equal sign in the pan balance activities they created on Scratch. Five eighth graders (4 girls and 1 boy) in Turkey participated in this study. Case study was employed as a qualitative research method. Data was gathered through the Scratch files created by the…

Fractions are foundational for success in algebra and STEM, but they are notoriously difficult for students. In past work, two separate branches of research from psychology and math education have studied fractions knowledge using different, isolated approaches. In this dissertation, I synthesize research from both fields about proportional…

This research is motivated by a linear equations system, which is the basis for studying necessary linear algebra materials, such as rank, range, linear independent/dependent, linear transformations, characteristic values and vectors. There are still prospective mathematics teachers who have difficulty solving linear equations system and…