Graded Troubleshooting (GTS) is a powerful routine that teachers can use easily to engender students' metacognitive thinking and boost their understanding of mathematics concepts and procedures. This article describes a new GTS activity designed to prompt students to efficiently exploit worked examples when asked to diagnose erroneous examples…

Developing both conceptual and procedural knowledge is important for students' mathematics competence. This study examined whether U.S. Grade 9 general education mathematics teachers' self-reported use of concept-focused instruction (CFI) and procedure-focused instruction (PFI) were associated differently with ninth graders' algebra achievement…

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…

This article focuses on first-year university students' understanding of concepts related to third-degree polynomial and trigonometric functions that they encountered during their study of Grade 12 mathematics. In this study three online questions from two first-year mathematics quizzes at the University of KwaZulu-Natal were analysed. The first…

Mental brackets constitute an idiosyncratic use of brackets sometimes used to evaluate arithmetic expressions and are closely connected with students' structure sense. The relevant literature describes the use of mental brackets focusing on primary school students and in the context of arithmetic. Using 181 high school students' solutions to seven…

Ben Zunica describes a lesson in which computational thinking has been successful in assisting students to understand the process of simplifying surds. The strengths and limitations of this approach are discussed. The author concludes that computational thinking can assist in solidifying understanding of a range of mathematical processes for…

This study examined "backward transfer," which we define as how students' ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for…

Making sense of algebraic expressions in context is multifaceted and extends beyond representing real-world situations using algebra. As part of Project MathTalk, the authors created videos that feature a pair of Grade 9 algebra 1 students coming to understand algebraic expressions in context. The students found three aspects of making sense of…

Different types of reasoning, such as intuitive, inductive, and deductive, are used in the generalization of figural patterns, as an important part of patterns in school mathematics. It is difficult to demarcate the constructive patterns where the regularity observed in the first few sentences is generalizable to the other sentences and each…

Textbooks are essential learning resources for both teachers and students, but they often contain misconceptions that can affect students' understanding of key concepts, particularly in biology. Misconceptions can arise from inaccurate or incomplete information, and they may lead to incorrect conceptual understanding. The clarity and accuracy of…

The present educational intervention study focused on addressing some areas of improvement in the teaching and learning of trigonometric ratios in high school students. A diagnostic evaluation was performed that revealed an instrumental understanding of trigonometric concepts by the students. Subsequently, a didactic sequence was implemented that…

This study examines the behavior of individuals when two intuitive rules in their minds lead to two different outcomes concerning a problem. In the absence of the formal knowledge, intuitive rules can affect the mathematical thinking. Hence, studies have usually compared the correct answer of the related formal knowledge with the incorrect one…

This study investigates the analogies used by a sample of secondary mathematics teachers when they described the concept of function. The study is concerned with understanding what analogies were used by the participant teachers and conceptions of functions positioned in those analogies. Using examples from a set of responses to five open-ended…

Quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students' difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old)…

Multiple representations can enhance students' understanding of mathematical concepts and complex information but can also pose well-documented challenges for students. Whereas instructional designs have been optimized to support students' "learning" with multiple representations, little is known about supportive "teaching"…