Backward transfer is defined as the influence that new learning has on individuals' prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students' prior ways of reasoning about linear functions. Two algebra classes and their teachers…

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…

College calculus plays an important role in STEM students' degree and career aspirations. One of the key factors considered in assessing a student's ability to be successful in calculus is their proficiency in topics from prior mathematics courses such as algebra and precalculus. This study set out to examine the impact of students' precalculus…

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…

Comparison is an important mechanism for learning in general, and comparing two worked examples has garnered support over the last 15 years as an effective tool for learning algebra in mainstream classrooms. This study was aimed at improving our understanding of how Modified for Language Support-Worked Example Pairs (MLS-WEPs) contribute to…

At a research university near the east coast, a college algebra class has been restructured into two large lectures a week, an active recitation size laboratory once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students work in groups on a…

Capitalizing on cognitive load theory and learning by analogy, we propose two instructional methods to learn a complex linear equation (e.g. two-step equation) by building on prior knowledge of a simpler linear equation (e.g. one-step equation). We will examine the proposal theoretically in this paper. In line with the design principles of…

This study examined the extent to which college algebra instructors employ pedagogical practices previously found to assist students master difficult STEM content material and address their own previous math deficits. Faculty classroom behaviors were assessed with a modified version of the Generalized Observation and Reflection Platform (GORP).…

This study aims to find out the dominant factors that cause students' difficulties in learning abstract algebra. The factors investigated in this study focused on attitude, motivation, prior knowledge, lecturer performance, learning methods, as well as teaching materials. This study is a descriptive research using a survey method. We involved 30…

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students' fraction number line acuity, rational number concepts and procedures.…

Faded worked examples have been used to promote problem solving performance, such as mathematics performance in many laboratory studies and short-term classroom studies. However, few studies have examined the ways educators may use fading in their own classroom on more accessible platforms that do not require programming experience. Further, few…

The study examines the impact of APOS theory on students' achievement in Elementary Linear Algebra (ELA) in accordance with their adequate prior knowledge (APK) and inadequate prior knowledge (IAPK). The mixed methods were used to determine the problem-solving test in ELA. Data were obtained from 65 industrial engineering students at Andalas…

Teacher knowledge is particularly important in middle school mathematics to enhance the meaningful transition from arithmetic to algebra. Developing a solid understanding of algebra requires teachers to unpack the meaning underlying the basic concepts, make clear and explicit distinctions among concepts, and facilitate students to connect these…

Mathematics is of major importance in science subjects. Unfortunately, students struggle with applying mathematics in science subjects, especially physics. In this qualitative study we demonstrate that transfer of algebraic skills from mathematics in physics class can be improved by using pre-knowledge effectively. We designed shift-problems…

In this study, we specify teacher's knowledge and beliefs that influence their enactment of mathematics tasks. The use of tasks in lessons requires teachers to consider students' mathematical thinking and to know how to effectively implement tasks. Among the many aspects of teachers' understanding of students and ability to implement tasks for…